The entropic force between two horizons of charged Gauss-Bonnet Black hole in de Sitter Spacetime
aa r X i v : . [ g r- q c ] J u l The entropic force between two horizons of chargedGauss-Bonnet Black hole in de Sitter Spacetime
Xiong-Ying Guo a,b , Ying Gao c , Huai-Fan Li a,b ∗ , Ren Zhao b a Department of Physics, Shanxi Datong University, Datong 037009, China b Institute of Theoretical Physics, Shanxi Datong University, Datong 037009, China b School of Mathematics and Statistics, Shanxi Datong University, Datong 037009, China
The basic equations of the thermodynamic system give the relationship betweenthe internal energy, entropy and volume of two neighboring equilibrium states. Byusing the functional relationship between the state parameters in the basic equation,we give the differential equation satisfied by the entropy of spacetime. We can obtainthe expression of the entropy by solving the differential equationy. This expressionis the sum of entropy corresponding to the two event horizons and the interactionterm. The interaction term is a function of the ratio of the locations of the black holehorizon and the cosmological horizon. The entropic force, which is strikingly similarto the Lennard-Jones force between particles, varies with the ratio of the two eventhorizons. The discovery of this phenomenon makes us realize that the entropic forcebetween the two horizons may be one of the candidates to promote the expansion ofthe universe.
PACS numbers: 04.70.Dy 05.70.Ce
I. INTRODUCTION
In the early period of inflation, our universe was in a quasi-de Sitter space. On the otherhand, with the inclusion of mysterious component with negative pressure, a large numberof dark energy models have been proposed to explain the cosmic acceleration. The simplestcandidate for dark energy is the cosmological constant (or vacuum energy density), withwhich our universe will naturally evolve into a new de Sitter phase. Finally, there has alsobeen flourishing interest in the duality relation of de Sitter space, promoted by the recent ∗ Email: [email protected]; [email protected](H.-F. Li) success of AdS/CFT correspondence in theoretical physics. Therefore, from observationaland theoretical point of view, it is rewarding to have a better understanding of the classicaland quantum properties of de Sitter space [1–23].One of the most promising modified gravity theories is the Gauss–Bonnet (GB) gravity(also referred to as Einstein–GB gravity), which offers the leading order correction to theEinstein gravity. The GB term α is exactly the second order term in the Lagrangian of themost general Lovelock gravity. Therefore, although α itself is quadratic in curvature tensors,the equations of gravitational fields are still of second order and naturally avoid ghosts. TheGB gravity possesses many important physical properties and has been heavily studied ingravitation and cosmology, also with emphasis in the extended phase space [24–42].In this paper, based on the fact that de Sitter spacetime satisfies the first law of ther-modynamics, the effective temperature and entropy of spacetime are obtained. Using therelationship between entropy and force [43–50], we discussed that the entropic force betweenthe black hole horizon and the cosmological horizon of charged Gauss-Bonnet black hole inde Sitter Spacetime(CGBDS). Since the entropy caused by the interaction between the twohorizons is composed of two parts, the entropic force between the two horizons is also com-posed of two parts. Some of them are proportional to the GB factor. The results show thatthe entropic force between the two horizons in CGBDS is not only related to the ratio ofthe location of the two horizons, but also related to the GB terms.By studying the entropic forces of the two parts of CGBDS with respect to the ratio ofthe locations of the event horizon, we find that they are strikingly similar to the Lennard-Jones force between particles with respect to the ratio of the coordinate locations of the twoparticles. When the two event horizons are close to each other, that is, when the spatialdistance between the two event horizons is small, the cosmic horizon accelerates away fromthe black holes horizon under the action of entropic force. When the locations of the twohorizons are relatively small, that is, the spatial distance between the two horizons is large,the separation speed of the two horizons slows down under the action of entropic force.The discovery of this phenomenon makes us realize that the entropy of interaction betweenthe event horizon of black hole and the cosmic horizon, and the entropic force generatedbetween the two event horizons, may be one of the alternatives to accelerate the expansionof the universe, that is, it may be a manifestation of dark energy. In particular, it should benoted that in our conclusion, the entropic force between the two horizons is proportional tothe GB factor, and the size of GB factor directly affects the entropic force between the twohorizons. If entropic force is one of the candidates to promote the expansion of the universe,then the value of GB factor directly affects the speed of the expansion of the universe.This paper in organized as follows. In Sec. II, we briefly introduce that the thermody-namic quantities corresponding to the black hole horizon and the cosmological horizon inCGBDS, and give the conditions that are satisfied when the black hole horizon and the cos-mic horizon have the same radiation temperature. In Sec. III, based on the condition thatCGBDS state parameter satisfies the first law of thermodynamics, we give that the expres-sion of the effective thermodynamic quantity of CGBDS system, and obtain the equivalenttemperature expression of CGBDS. Moreover, we find the differential equation of the en-tropy of CGBDS system, and obtain the interaction term of entropy of CGBDS system bysolving the differential equation. In Sec. IV, we discussed the entropic force between thetwo horizons in CGBDS by using the entropic force relationship, and obtained the entropyexpression between the two horizons. The curves of the entropic force with respect to theratio of the two horizons are compared with the Lennard-Jones force with respect to theratio of the coordinates of the two particles. The Sec. V is a discussion and summary. Forsimplicity, we adopt the units ~ = c = k B = G = 1 in this paper. II. CHARGED GAUSS-BONNET BLACK HOLE IN DE SITTER SPACETIME
Higher derivative curvature terms occur in many occasions, such as in the semiclassicallyquantum gravity and in the effective low-energy action of superstring theories. Among themany theories of gravity with higher derivative curvature terms, due to the special featuresthe Gauss-Bonnet gravity has attract much interest. The thermodynamic properties andphase structures of GB-AdS black hole have been briefly discussed in [35]. In Refs. [30, 38],the critical phenomena and phase transition of the charged GB-AdS black hole have beenstudied extensively. In this paper, we study the thermal properties of charged GB-dS blackhole after considering the connections between the black hole horizon and the cosmologicalhorizon.The action of d -dimensional Einstein-Gauss-Bonnet-Maxwell theory with a bare cosmo-logical constant Λ reads I = 116 π Z d d x √− g (cid:2) R −
2Λ + α ( R µνγδ R µνγδ − R µν R µν + R ) − πF µν F µν (cid:3) , (2.1)where the GB coupling α has dimension [length] and can be identified with the inversestring tension with positive value if the theory is incorporated in string theory, thus we shallconsider only the case α > F µν is the Maxwell field strength defined as F µν = ∂ µ A µ − ∂ ν A ν with vector potential A µ . In addition, let us mention here that the GB term is a topologicalterm in d = 4 dimensions and has no dynamics in this case. Therefore we will consider d ≥ ds = − f ( r ) dt + f − ( r ) dr + r h ij dx i dx j , (2.2)where h ij dx i dx j represent the line of a d − d − d − k and volume Σ k . Without loss of the generality, onemay take k = 1, 0 and −
1, corresponding to the spherical, Ricci fiat and hyperbolic topologyof the black hole horizon, respectively. The metric function f ( r ) is given by [51–53] f ( r ) = k + r α " − s π ˜ αM ( d − k r d − − αQ ( d − d − r d − + 8 ˜ α Λ( d − d − , (2.3)where ˜ α = ( d − d − α , M and Q are the mass and charge of black hole respectively,and pressure P P = − Λ8 π = − ( d − d − πl . (2.4)Note that in order to have a well-defined vacuum solution with M = Q = 0, the effectiveGauss-Bonnet coefficient ˜ α and pressure P have to satisfy the following constraint64 π ˜ αP ( d − d − ≤ . (2.5)When d = 5, the location of the event horizon of the black hole r + and the location ofthe cosmic event horizon r c satisfy the relation f ( r + ,c ) = 0. The equations f ( r + ) = 0 and f ( r c ) = 0 are rearranged to M = 3Σ k r π (cid:18) k + k ˜ αr (cid:19) − Σ k r Λ32 π + Σ k Q πr (2.6) M = 3Σ k r c π (cid:18) k + k ˜ αr c (cid:19) − Σ k r c Λ32 π + Σ k Q πr c . (2.7)From Eqs. (2.6) and (2.7), we can obtain M = 3Σ k kr c x π (1 + x ) + 3Σ k k ˜ α π + Σ k Q πr c (1 + x ) x (cid:0) x + x (cid:1) , (2.8)Λ = 6 r c (1 − x ) k (1 − x ) − Q (1 − x )2 r c x (1 − x ) , (2.9)where x = r + /r c . From Eqs. (2.3), (2.8) and (2.9), we can obtain f ′ ( r + ) = 2 kr + (1 − x )( r + 2 ˜ αk )(1 + x ) − Q [(1 + x ) − x ]6 r ( r + 2 ˜ αk )(1 + x )= 2 kr c x (1 − x )( r c x + 2 ˜ αk )(1 + x ) − Q [(1 + x ) − x ]6( r c x + 2 ˜ αk ) r c x (1 + x ) , (2.10) f ′ ( r c ) = − kr c (1 − x )( r c + 2 ˜ αk )(1 + x ) − Q [ x (1 + x ) − r c x ( r c + 2 ˜ αk )(1 + x )= − kr + x (1 − x )( r + 2 ˜ αkx )(1 + x ) − x Q ( x (1 + x ) − r ( r + 2 ˜ αkx )(1 + x ) . (2.11)Some thermodynamic quantities associated with the cosmological horizon are T c = − f ′ ( r c )4 π S c = Σ k r c (cid:18) αkr c (cid:19) , Φ c = Σ k r c T c , S c and Φ c denote the Hawking temperature, the entropy and the charged potential. Forthe black hole horizon, associated thermodynamic quantities are T + = f ′ ( r + )4 π , S + = Σ k r c x (cid:18) α c kr c x (cid:19) , Φ + = Σ k r c x . (2.13)From Eqs. (2.1) and (2.11), we found that the charge Q of spacetime meets Q = 12 kr c (1 + x )( r c x − αk ) x [ r c (1 + x )(1 + x + 3 x + x + x ) + 2 ˜ αk (1 + x )]= 12 kr (1 + x )( r − x ˜ αk )[ r (1 + x )(1 + x + 3 x + x + x ) + 2 x ˜ αk (1 + x )] , (2.14)the radiation temperature at the event horizon of the black hole is the same as that at thecosmic event horizon, T = T + = T c = kr c (1 + x ) (1 − x )2 π [ r c (1 + x )(1 + x + 3 x + x + x ) + 2 ˜ αk (1 + x )]= kr + x (1 + x ) (1 − x )2 π ( r (1 + x )(1 + x + 3 x + x + x ) + 2 x ˜ αk (1 + x )) . (2.15) III. THE EFFECTIVE THERMODYNAMICS QUANTITES OF BLACK HOLE
Considering the relation between the black hole horizon and the cosmological horizon, wecan derive the effective thermodynamic quantities and corresponding first law of black holethermodynamics dM = T eff dS − P eff dV + Φ eff dQ, (3.1)here the thermodynamic volume is that between the black hole horizon and the cosmologicalhorizon, namely [22, 23, 54] V = V c − V + = Σ k r c (cid:0) − x (cid:1) . (3.2)Considering the expressions of entropy corresponding to the two horizons, as well as thedimensions and ˜ αk terms, we assume that the total entropy of spacetime is S = Σ k r c (cid:20) f ( x ) + 6 ˜ αkr c f ( x ) (cid:21) , (3.3)here the function f ( x ) and f ( x ) represents the extra contribution from the correlations ofthe two horizons. Taking Q , ˜ α as constant, substituting Eqs. (2.8), (3.2) and (3.3) into Eq.(3.1), we can obtain the effective temperature T eff of the system T eff = r c kx (1 − x + x ) − Q r c x (1 + x − x + x + x )2 π (1 + x ) { r c [ f ′ ( x )(1 − x ) + 3 x f ( x )] + 6 ˜ αk [ f ′ ( x )(1 − x ) + x f ( x )] } . (3.4)When Q satisfies equation (2.14), the temperature corresponding to the two horizons isequal. In this case, we believe that the effective temperature of spacetime should also beradiation temperature, and T eff = kr c (1 + x ) (1 − x )2 π [ r c (1 + x )(1 + x + 3 x + x + x ) + 2 ˜ αk (1 + x )] . (3.5)From Eqs.(3.4), (2.14) and Eq. (3.5), we can obtainΣ k r c k (1 − x ) (1 + x )16 π (1 + x ) ( r c x + 2 ˜ αk )( r c + 2 ˜ αk ) x + Σ k r c k ˜ α (1 − x ) (1 + x )8 π (1 + x ) ( r c x + 2 ˜ αk )( r c + 2 ˜ αk ) x = 14 πr c kr c (1 − x ) (1 − x )3 x (1 + x ) ( r c + 2 ˜ αk )( r c x + 2 ˜ αk ) × (cid:20) Σ k r c [ f ′ ( x )(1 − x ) + 3 x f ( x )] + 3Σ k r c ˜ αk [ f ′ ( x )(1 − x ) + x f ( x )] (cid:21) . (3.6)Since ˜ αk in Eq. (3.6) is an independent variable, the same terms at both sides of Eq. (3.6)should be equal, i.e (1 − x ) f ′ ( x ) + 3 x f ( x ) = 3 x (1 + x )(1 − x ) , (1 − x ) f ′ ( x ) + x f ( x ) = (1 + x )(1 − x ) . (3.7)Substituting Eq. (3.7) into Eq. (3.4) to obtain the effective temperature of spacetime T eff = r c kx (1 − x + 2 x − x ) − Q r c x (1 − x + 2 x − x )2 π [ r c x (1 + x ) + 2 ˜ αk (1 + x )]= r + k (1 − x + 2 x − x ) − Q (1 − x + 2 x − x ) / (12 r )2 π [ r (1 + x ) + 2 ˜ αk (1 + x )] . (3.8)When k = 1, r + = 1, ˜ αk = 0 .
05 , from Eqs.(2.12), (2.13), (2.15) and (3.8), we can plot thecurve T + ,c − x , T − x and T eff − x in Fig. 1. From Fig.1, we can see that the intersection of T + T c TT eff T + (cid:144) T c (cid:144) T (cid:144) T eff Q = T + T c TT eff T + (cid:144) T c (cid:144) T (cid:144) T eff Q = T + T c TT eff T + (cid:144) T c (cid:144) T (cid:144) T eff Q = T + T c TT eff T + (cid:144) T c (cid:144) T (cid:144) T eff Q = FIG. 1: (Color online) The T − x diagram for Charged Gauss-Bonnet black holes in de Sitter spacetime. curves T + ,c − x , T − x and T eff − x increases with the decrease of Q , and the intersectionof the curve approaches x →
1. When the initial conditions satisfy f (0) = 1, f (0) = 1, thesolution of Eq. (3.7) is f ( x ) = 117 (1 − x ) / − x ) − x (1 + x )7(1 − x )= 117 (1 − x ) / − x ) + 7(1 − x − x )7(1 − x ) + 1 + x = ˜ f ( x ) + 1 + x , (3.9) f ( x ) = 9(1 − x ) / − (4 − x − x + 4 x )5(1 − x )= 9(1 − x ) / − (9 − x − x )5(1 − x ) + 1 + x = ˜ f ( x ) + 1 + x. (3.10)From Eqs.(2.12), (2.13), (3.3), (3.9) and (3.10), we can obtain that the entropy of spacetimeincludes not only the sum of entropy S + + S c corresponding to the event horizon of the blackhole and the cosmic event horizon, but also the entropy ˜ S + ˜ S caused by the interaction ofthe two event horizons ˜ S = Σ k r c ˜ f ( x ) , ˜ S = 3Σ k αkr c ˜ f ( x ) . (3.11)From Eq.(3.11), we can plot ˜ f ( x ) − x , ˜ f ( x ) − x , 0 < x ≤ S Ž H x L S Ž H x L - - S Ž H x L(cid:144) S Ž H x L FIG. 2: (Color online) The entropy from the interaction between horizons of black holes and our universe. In the calcluate,we set k ˜ α = 1, r c = 1. from Fig.2, the two curves have the same change rule, and the amplitude of ˜ S curve isproportional to ˜ α .From Eq.(3.1), we can obtain the effective pressure P eff and the effective potential Φ eff P eff = − (cid:18) ∂M∂V (cid:19) Q,S = − (cid:0) ∂M∂x (cid:1) r c (cid:16) ∂S∂r c (cid:17) x − (cid:16) ∂M∂r c (cid:17) x (cid:0) ∂S∂x (cid:1) r c (cid:0) ∂V∂x (cid:1) r c (cid:16) ∂S∂r c (cid:17) x − (cid:16) ∂V∂r c (cid:17) x (cid:0) ∂S∂x (cid:1) r c = (1 − x ) (cid:2) [ f ( x )] + 6 ˜ αkx /r f ( x ) (cid:3) π (1 + x ) [ r (1 + x ) + 2 ˜ αk (1 + x )] (cid:20) kx − Q x (1 + 2 x )4 r (cid:21) − (1 − x ) [ f ′ ( x ) + 6 ˜ αk/r c f ′ ( x )]24 π [ r (1 + x ) + 2 ˜ αk (1 + x )] (cid:20) kx − Q x r (cid:0) x + x (cid:1)(cid:21) , (3.12)Φ eff = (cid:18) ∂M∂Q (cid:19) S,V = Σ k Q πr (1 + x ) (cid:0) x + x (cid:1) . (3.13) IV. THE ENTROPIC FORCE OF TWO HORIZON OF GBDST
The entropic force of the thermodynamic system is expressed as [43, 46–50] F = − T ∂S∂r , (4.1)where T is the temperature of system, r = r c − r + = r c (1 − x ). We consider the entropic forcebetween the event horizon of the black hole and the event horizon of the universe. FromEq.(3.11), we know that the entropy caused by the interaction between the event horizon ofthe black hole and the cosmic event horizon is˜ S = Σ k r c ˜ f ( x ) , ˜ S = 3Σ k αkr c ˜ f ( x ) . (4.2)According to the entropic force relation Eq.(4.1), we can obtain the entropic force of inter-action between the two horizons can be expressed as F = − T eff ∂ ( ˜ S + ˜ S ) ∂r ! T eff , (4.3)where T eff is the effective temperature of the system. From Eq.(4.3), we can obtain F ( x ) = T eff (cid:16) ∂ ( ˜ S + ˜ S ) ∂r c (cid:17) x (cid:16) ∂T eff ∂x (cid:17) r c − (cid:16) ∂ ( ˜ S + ˜ S ) ∂x (cid:17) r c (cid:16) ∂T eff ∂r c (cid:17) x (1 − x ) (cid:16) ∂T eff ∂x (cid:17) r c + r c (cid:16) ∂T eff ∂r c (cid:17) x = ˜ F ( x ) + ˜ F ( x ) , (4.4)where ˜ F ( x ) = T eff (cid:16) ∂ ˜ S∂r c (cid:17) x (cid:16) ∂T eff ∂x (cid:17) r c − (cid:16) ∂ ˜ S∂x (cid:17) r c (cid:16) ∂T eff ∂r c (cid:17) x (1 − x ) (cid:16) ∂T eff ∂x (cid:17) r c + r c (cid:16) ∂T eff ∂r c (cid:17) x , ˜ F ( x ) = T eff (cid:16) ∂ ˜ S ∂r c (cid:17) x (cid:16) ∂T eff ∂x (cid:17) r c − (cid:16) ∂ ˜ S ∂x (cid:17) r c (cid:16) ∂T eff ∂r c (cid:17) x (1 − x ) (cid:16) ∂T eff ∂x (cid:17) r c + r c (cid:16) ∂T eff ∂r c (cid:17) x . (4.5)The interaction between the two horizons is divided into two parts, ˜ F ( x ) and ˜ F ( x ), where˜ F ( x ) is caused by ˜ S , and ˜ F ( x ) is caused by ˜ S . Since ˜ S is proportional to GB factor ˜ α ,˜ F ( x ) is proportional to ˜ α . ˜ F ( x ) is greatly affected by GB factor ˜ α , when ˜ α →
0, ˜ F ( x ) → x between the two horizons, and the influence of different parameters on the entropicforce F ( x ) between the two horizons, so we can plot ˜ F ( x ) − x and ˜ F ( x ) − x for differentparameters. From Fig. 3, we find that the variation curves of ˜ F ( x ) − x and ˜ F ( x ) − x with0 Α= Α= Α= F Ž H x L Α= Α= Α= - F Ž H x L Α= Α= Α= - H x L Q = = = - F Ž H x L Q = = = F Ž H x L Q = = = - H x L FIG. 3: (Color online) The entropic force changes with the ratio of the radius of the event horizon of the black hole to theradius of the cosmic event horizon for charged Gauss-Bonnet Black hole in de Sitter Spacetime with different Gauss-Bonnetfactor ˜ α and charged Q . respect to the location ratio of the two horizons are similar and have the same variationrules, and the amplitude of the curves is proportional to ˜ α .From Fig.3, the general change of entropic force with respect to x is that as x →
1, theentropic force goes to infinity. It is shown that when the event horizon of the black holein de Sitter is close to the cosmic event horizon, the two event horizons are affected bythe infinite entropic force, which accelerates the separation between the two event horizons.This corresponds to what we now think of as the beginning of the universe’s explosion, whenthe expansion of the universe accelerated. With the separation of the two horizons, namelythe value of x is reduced, the entropic force between two horizon decreases, and when the x = x , the first time fellowship with the x axis, the entropic force between the two horizonis zero, between the event horizon is not affected by external force, to maintain state ofseparation between two horizon, when the x continue to reduce the negative entropic forcebetween the two horizons, in the interval between two horizon deceleration separation. Thiscase corresponds to our universe is slowing inflation. With the decrease of x , that is, theseparation between the two horizons, the entropic force between the two horizons continuesto decrease, and the entropic force curve gradually approaches the horizontal axis, and theentropic force between the two horizons approaches zero.Comparing Fig.3 with the curve of Lennard-Jones forces between two particles as theyvary in location [55, 56], we find that the curves obtained by completely different methodsare so strikingly similar that the Lennard-Jones force of the two particles is intrinsically1related to the entropic force between the two event horizons. Since the expansion of theuniverse is affected by various substances in the universe, the curve F ( x ) − x in Fig.3 reflectsthe influence of different parameters of space-time on the entropic force between the twohorizons. If the entropic force is one of the internal forces driving the expansion of theuniverse, Fig.3 shows that the accelerated expansion of our universe is influenced by variousparameters. In particular, under the same parameters, the amplitude of the curve F ( x ) − x is proportional to ˜ α , indicating that the force between the two horizons is proportional tothe GB factor, so the size of the GB factor ˜ α is proportional to the expansion rate of theuniverse. V. DISCUSSION AND SUMMARY
According to the discussion in the section (IV), comparing the entropic force curve ˜ F ( x ) − x and ˜ F ( x ) − x of the location ratio between the two horizons in GB space-time given in Fig.3with the Lennard-Jones force changing with the location curve given in literature [55, 56],we find that the two curves obtained in different ways are very similar. The entropic forcerelation between the two horizons is derived from the theory within the framework of generalrelativity. It is derived from the theory of general relativity in combination with quantummechanics and thermodynamics. The Lennard-Jones force between the two particles is basedon the experiment. Although the method used is completely different, the results obtainedby the two particles are surprisingly similar. This conclusion indicates that the entropicforce between the two event horizons is related to the Lennard-Jones force between the twoparticles. In particular, the entropic force between the two horizons is proportional to GBfactor ˜ α , and the size of ˜ α plays a direct role in the acceleration between the two horizons.If entropic force is one of the kinetic energy driving the expansion of the universe, then thesize of GB factor ˜ α is directly related to the acceleration of the expansion of the universe.In the framework of general relativity, the entropic force of the interaction between theevent horizon of a black hole and the event horizon of the universe deduced by theory has avery high similarity to the Lennard-Jones force between two particles verified by experiment.Therefore, the conclusion we give reveals the internal relationship between general relativity,quantum mechanics and thermodynamics. This analogy provides a new way for us to studythe interaction between particles and the microscopic states of particles inside black hole,2as well as the relationship between Lennard-Jones potential and the microscopic states ofparticles inside a normal thermodynamic system. Acknowledgements
We would like to thank Prof. Zong-Hong Zhu and Meng-Sen Ma for their indispensablediscussions and comments. This work was supported by the Young Scientists Fund of theNational Natural Science Foundation of China (Grant No.11205097), in part by the Na-tional Natural Science Foundation of China (Grant No.11475108), Supported by Programfor the Natural Science Foundation of Shanxi Province, China(Grant No.201901D111315)and the Natural Science Foundation for Young Scientists of Shanxi Province,China (GrantNo.201901D211441). [1] R.G. Cai,
Cardy-Verlinde formula and thermodynamics of black holes in de Sitter spaces . Nucl.Phys. B628 (2002) 375; [hep-th/0112253][2] S. Mbarek, R. B. Mann,
Reverse Hawking-Page Phase Transition in de Sitter Black Holes .JHEP02(2019)103; 1808.03349 [hep-th][3] F. Simovic, R. B. Mann,
Critical Phenomena of Born-Infeld-de Sitter Black Holes in Cavities .JHEP05(2019)136 ; arXiv:1904.04871[gr-qc][4] F. Simovic and R. Mann,
Critical Phenomena of Charged de Sitter Black Holes in Cavities
Classical and Quantum Gravity,Volume 36 Number 1 (2018) 014002; 1807.11875 [gr-qc][5] B. P. Dolan, D. Kastor, D. Kubiznak, R. B. Mann, J. Traschen,
Thermodynamic Volumesand Isoperimetric Inequalities for de Sitter Black Holes , Phys. Rev. D 87, 104017 (2013),arXiv:1301.5926 [hep-th][6] Y. Sekiwa,
Thermodynamics of de Sitter black holes: Thermal cosmological constant . Phys.Rev. D 73(2006)084009; hep-th/0602269[7] M. Urano and A. Tomimatsu,
Mechanical First Law of Black Hole Spacetimes with Cosmolog-ical Constant and Its Application to Schwarzschild-de Sitter Spacetime . Class. Quant. Grav.26(2009)105010 ;arXiv:0903.4230[gr-qc][8] M. Azreg-Aiou,
Charged de Sitter-like black holes: quintessence-dependent enthalpy and new extreme solutions , Eur. Phys. J. C75 (2015) 34, [1410.1737].[9] D. Kubiznak and F. Simovic, Thermodynamics of horizons: de Sitter black holes and reen-trant phase transitions , Class. Quant. Grav. 33 (2016) 245001;arXiv:1507.08630 [hep-th]
Cosmography of KNdS Black Holes and Isen-tropic Phase Transitions , Class. Quant. Grav. 33(2016) 105007; arXiv:1509.02343 [hep-th][11] S. Bhattacharya, A. Lahiri,
Mass function and particle creation in Schwarzschild-de Sitterspacetime , Eur. Phys. J. C (2013) 73:2673; arXiv:1301.4532 [gr-qc][12] L.C. Zhang and R. Zhao,
The critical phenomena of Schwarzschild-de Sitter black hole , Euro-phys. Lett. 113 (2016) 10008[13] R.G. Cai,
Cardy-Verlinde formula and asymptotically de Sitter spaces , Phys. Lett. B525 (2002)33; arXiv: hep-th/0111093[14] T. Pappas, P. Kanti, and N. Pappas,
Hawking radiation spectra for scalar fields by a higher-dimensional Schwarzschild–de Sitter black hole , Phys. Rev. D 94, 024035 (2016)[15] P. Kanti and T. Pappas,
Effective Temperatures and Radiation Spectra for a Higher-Dimensional Schwarzschild-de-Sitter Black-Hole , Phys. Rev. D 96, 024038 (2017) 1705.09108[16] S. Bhattacharya,
A note on entropy of de Sitter black holes , Eur. Phys. J. C76 (2016)112;arXiv:1506.07809 [gr-qc][17] L.C. Zhang, R. Zhao, M.S. Ma,
Entropy of Reissner–Nordstr¨om–de Sitter black hole . PhysicsLetters B 761 (2016) 74–76; arXiv:1610.09886 [gr-qc][18] L.C. Zhang, R. Zhao,
The entropy force in Reissner–Nordstr¨om–de Sitter black hole . PhysicsLetters B 797 (2019) 134798[19] Y.B. Ma, Y. Zhang, L.C. Zhang, L. Wu, Y.m. Huang, Y. Pan,
Thermodynamic properties ofhigher-dimensional dS black holes in dRGT massive gravity . Eur. Phys. J. C80, 213(2020);arXiv:2003.05483 [hep-th][20] J. Dinsmore, P. Draper, D. Kastor, Y. Qiu, J. Traschen,
Schottky Anomaly of de Sitter BlackHoles , [arXiv:1907.00248 [hep-th]][21] M. Chabab, H. El Moumni, J. Khalloufi,
On Einstein-non linear-Maxwell-Yukawa de-Sitterblack hole thermodynamics . arXiv:2001.01134 [hep-th][22] X.Y. Guo, H.F. Li, L.C. Zhang, R. Zhao,
Thermodynamics and phase transition of in theKerr-de Sitter black hole , Physical Review D, 2015.91. 084009 [23] H.H. Zhao , M. S. Ma, L.C. Zhang and R. Zhao, P - V criticality of higher dimensional chargedtopological dilaton de Sitter black holes , Phys. Rev. D,90, 064018(2014)[24] B. Chen and P.C. Li,
Static Gauss-Bonnet Black Holes at Large D . 10.1007/JHEP05(2017)025;arXiv:1703.06381v1 [hep-th][25] D. D. Doneva, K. V. Staykov, S. S. Yazadjiev,
Gauss-Bonnet black holes with a massive scalarfield . Phys. Rev. D 99, 104045 (2019); arXiv:1903.08119v2 [gr-qc][26] S. H. Hendi, S. Panahiyan, and B. E. Panah,
Charged black hole solutions in Gauss-Bonnet-Massive Gravity , J. High Energy Phys. 1601, 129 (2016),arXiv:1507.06563[hep-th].[27] C.Y. Zhang, P.C. Li, M.Y. Guo,
Greybody factor and power spectra of the Hawking radiationin the novel 4D Einstein-Gauss-Bonnet de-Sitter gravity . arXiv:2003.13068v1 [hep-th][28] R. A. Konoplya, A. Zhidenko,
Long life of Gauss-Bonnet corrected black holes . Phys. Rev.D.82.084003 (2010)[29] B. Chen, Z.Y. Fan, P.C. Li and W.C. Ye,
Quasinormal modes of Gauss-Bonnet black holes atlarge D . JHEP01(2016)085. 1511.08706[30] R.G. Cai, L.M. Cao, L. Li, and R.Q. Yang,
P-V criticality in the extended phase space ofGauss-Bonnet black holes in AdS space
Thermodynamic geometry of novel 4-D Gauss Bonnet AdS Black Hole .arXiv:2003.13382v1 [gr-qc][32] H. Witek, L. Gualtieri, and P. Pani,
Towards numerical relativity in scalar Gauss-Bonnetgravity: decomposition beyond the small-coupling limit . arXiv:2004.00009 [gr-qc][33] S.W. Wei, and Y.X. Liu,
Testing the nature of Gauss-Bonnet gravity by four-dimensionalrotating black hole Shadow . arXiv:2003.07769v2 [gr-qc][34] Y.Y. Wang, B.Y. Su, and N. Li,
The Hawking–Page phase transitions in the extended phasespace in the Gauss–Bonnet gravity . arXiv:1905.07155v1 [gr-qc][35] R.G. Cai,
Gauss-Bonnet black holes in AdS spaces , Phys. Rev. D 65, 084014 (2002),arXiv:hep-th/0109133[hep-th].[36] P. Kanti, B. Kleihaus, and J. Kunz,
Wormholes in dilatonic Einstein-Gauss-Bonnet theory ,Phys. Rev. Lett. 107, 271101 (2011), arXiv:1108.3003[gr-qc].[37] D. D. Doneva and S. S. Yazadjiev,
New Gauss-Bonnet black holes with curvature-inducedscalarization in extended scalar-tensor theories , Phys. Rev. Lett. 120, 131103 (2018),arXiv:1711.01187[gr-qc]. [38] S.W. Wei and Y.X. Liu, Critical phenomena and thermodynamic geometry of charged Gauss-Bonnet AdS black holes , Phys. Rev. D 87, 044014 (2013), arXiv:1209.1707[gr-qc].[39] W. Xu, H. Xu and L. Zhao,
Gauss-Bonnet coupling constant as a free thermodynamical variableand the associated criticality , Eur. Phys. J. C 74, 2970 (2014), arXiv:1311.3053[gr-qc].[40] D.C. Zou, Y. Liu, and B. Wang,
Critical behavior of charged Gauss-Bonnet AdS black holesin the grand canonical ensemble , Phys. Rev. D 90, 044063 (2014), arXiv:1404.5194[hep-th].[41] Y.G. Miao and Z.M. Xu,
Parametric phase transition for a Gauss-Bonnet AdS black hole ,Phys. Rev. D 98, 084051 (2018), arXiv:1806.10393[hep-th].[42] B. Kleihaus, J. Kunz, and E. Radu,
Rotating black holes in dilatonic Einstein-Gauss-Bonnettheory , Phys. Rev. Lett. 106, 151104 (2011), arXiv:1101.2868[gr-qc].[43] E. Verlinde,
On the Origin of Gravity and the Laws of Newton . arXiv:1001.0785 [hep-th];JHEP 04 (2011) 029.[44] C. P. Panos, Ch. C. Moustakidis,
A simple link of information entropy of quantum and classicalsystems with Newtonian r dependence of Verlinde’s entropic force , Physica A 518, 384 (2019);arXiv:1809.09484[45] L. Calderon, M. T. Martin, A. Plastino, M. C. Rocca, V. Vampa, Relativistic treatment ofVerlinde’s emergent force in Tsallis’ statistics . Modern Physics Letters A, 34, 1950075 (2019)arXiv:1903.08150 [cond-mat.stat-mech][46] N. Komatsu,
Generalized thermodynamic constraints on holographic-principle-based cosmolog-ical scenarios . Phys. Rev. D 99, 043523 (2019); arXiv:1810.11138 [gr-qc][47] N. Komatsu,
Thermodynamic constraints on a varying cosmological-constant-like term fromthe holographic equipartition law with a power-law corrected entropy . Phys. Rev. D 96, 103507(2017); arXiv:1707.09101 [gr-qc][48] N. Komatsu, S. Kimura, S. Kimura,
General form of entropy on the horizon of the universein entropic cosmology . Phys. Rev. D 93, 043530 (2016); arXiv:1511.04364 [gr-qc][49] Y.F. Cai, E. N. Saridakis,
Inflation in entropic cosmology: Primordial perturbations and non-Gaussianities . Physics Letters B 697(2011)280[50] Y.F. Cai, J. Liu, H. Li,
Entropic cosmology: A unified model of inflation and late-time accel-eration . Phys. Lett B690 (2010)213; arXiv: 1003.4526 [astro-ph.CO][51] W. Xu, H. Xu, L. Zhao,
Gauss-Bonnet coupling constant as a free thermodynamical variableand the associated criticality , Eur. Phys. J. C (2014) 74:2970. [52] S. W. Wei, Y. X. Liu, Triple points and phase diagrams in the extended phase space of chargedGauss-Bonnet black holes in AdS space , Phys. Rev. D90, 044057 (2014).[53] M.S. Ma, L.C. Zhang, H.H Zhao, and R. Zhao,
Phase Transition of the Higher DimensionalCharged Gauss-Bonnet Black Hole in de Sitter Spacetime . Advances in High Energy Physics.Volume 2015, Article ID 134815, 8 pages 1410.5950[54] B. P. Dolan, D. Kastor, D. Kubiznak, R. B. Mann, J. Traschen,
Thermodynamic Volumes andIsoperimetric Inequalities for de Sitter Black Holes , arXiv:1301.5926[55] Y.G. Miao and Z.M. Xu,
On thermal molecular potential among micromolecules in chargedAdS black hole , Phys. Rev. D 98, 044001 (2018);arXiv:1712.00545[hep-th][56] Y.G. Miao and Z.M. Xu,