Phase transition of the higher dimensional charged Gauss-Bonnet black hole in de Sitter spacetime

We study the phase transition of charged Gauss-Bonnet-de Sitter (GB-dS) black hole. For black holes in de Sitter spacetime, there is not only black hole horizon, but also the cosmological horizon. The thermodynamic quantities on the both horizons satisfy the first law of the black hole thermodynamics, respectively; moreover, there are additional connections between them. Using the effective temperature approach, we obtained the effective thermodynamic quantities of charged GB-dS black hole. According to Ehrenfest classification, we calculate some response functions and plot their figures, from which one can see that the spacetime undergoes a second-order phase transition at the critical point. It is shown that the critical values of effective temperature and pressure decrease with the increase of the value of GB parameter $\alpha$.


I. INTRODUCTION
Due to the existence of Hawking radiation and the entropy, black holes may display themselves like thermodynamic systems [1][2][3][4]. Phase transitions and critical phenomena are important characteristics of ordinary thermodynamic system. Thus, the natural question to ask is whether there also exists phase transition in the black hole thermodynamics. The pioneer work of Hawking and Page give us an definitive answer [5]. Recently, the phase transitions of black holes in asymptotically anti de-Sitter (AdS) spacetime have received considerable attention [6][7][8][9][10][11][12][13][14][15]. In particular, by considering the cosmological constant as thermodynamic pressure P = − Λ 8π = (d−1)(d−2) 16πl 2 , one can introduce an extended phase space, in which the P − V criticality has been extensively discussed [16-29, 31, 36]. It is shown that many black holes in AdS spacetime exhibit similar critical behaviors to that of Van der Waals liquid-gas system.
The astronomical observations show that our Universe is probably an asymptotically de Sitter (dS) one. It raises the interest on black holes in dS spacetime. There are many works on the thermal properties of black holes in de Sitter(dS) spacetime [32][33][34][35][36][37][38][39][40][41][42][43]. As is well known, there are multi-horizons for black holes in dS spacetime. On the black hole event horizon and the cosmological horizon there are different temperatures, which prevent the black hole thermodynamic system in equilibrium. It should be noted that there are common parameters M , Q, Λ in the thermodynamic quantities corresponding to the black hole horizon and that corresponding to the cosmological horizon. Thus, these thermodynamic quantities are not independent. Taking into account their connections is relevant to the understanding of thermal properties of dS spacetime.
Higher derivative curvature terms occur in many occasions, such as in the semiclassically quantum gravity and in the effective low-energy action of superstring theories. Among the many theories of gravity with higher derivative curvature terms, due to the special features the Gauss-Bonnet (GB) gravity has attract much interest. The thermodynamic properties and phase structures of GB-AdS black hole have been briefly discussed in [44]. In Refs. [23][24][25], the critical phenomena and phase transition of the charged GB-AdS black hole have been studied extensively.
In this paper, we study the thermal properties of charged GB-dS black hole after considering the connections between the black hole horizon and the cosmological horizon. Using the effective equilibrium temperature approach, we calculate some effective thermodynamic quantities, from which we can analyze whether there exists phase transition for charged GB-dS black hole.
The paper is arranged as follows: in the next section we simply introduce the charged Gauss-Bonnet-dS black hole. In section 3, we will calculate the effective thermodynamic quantities. In section 4, we derive the heat capacity at constant pressure, the volume expansivity β, and the isothermal compressibility κ and plot some curves to demonstrate the phase transition. We will make some concluding remarks in section 5. (we use the units G d =h = c = k B = 1)

II. CHARGED GAUSS-BONNET-DS BLACK HOLE
The action of the d-dimensional Einstein-Gauss-Bonnet-Maxwell-dS theory has the form where α is the Gauss-Bonnet coupling constant and the cosmological constant is Λ = for dS spacetime, F µν is the Maxwell field strength. The d-dimensional static charged Gauss-Bonnet-dS black hole solution for the action is described by with the metric function given by [23,24,29] f (r) = 1 + by eliminating the Λ, and The surface gravities of black hole horizon and the cosmological horizon are The thermodynamic quantities for the two horizons satisfy the first law of black hole thermodyamic [33,36] where

III. EFFECTIVE THERMODYNAMIC QUANTITIES OF CHARGED GAUSS-BONNET BLACK HOLE
We have given the thermodynamic quantities without considering the relationship between the black hole horizon and the cosmological horizon. Because there are four variables M , Q , Λ andα in the spacetime, the thermodynamic quantities corresponding to the black hole horizon and the cosmological horizon are functions of these variables. Through the four variables, there are some connections between the thermodynamic quantities corresponding to the black hole horizon and the ones corresponding to the cosmological horizon. When the thermodynamic property of charged de Sitter spacetime is studied, we must consider the relationship with the two horizon. Recently, by studying Hawking radiation of de Sitter spacetime, refs, [45][46][47] obtained that the outgoing rate of the charged de Sitter spacetime which radiates particles with energy ω is where ∆S + and ∆S c are Bekenstein-Hawking entropy difference corresponding to the black hole horizon and the cosmological horizon after the charged de Sitter spacetime radiates particles with energy ω. Therefore, the thermodynamic entropy of the charged de Sitter spacetime is the sum of the black hole horizon entropy and the cosmological horizon entropy: Recently, thermodynamic volume of charged dS black hole is given by [33,36] From (8), one can obtain Substituting (11) and (12) into (13), we obtain , dS c = r 3 + (r 2 c + 2α)dS r 3 c (r 2 + + 2α) + r 3 + (r 2 c + 2α) Substituting (14) into (7) and (8), we get where the effective temperature T ef f , effective electric potential Φ ef f and effective pressure P ef f are respectively , Substituting (5), (6) and (9) into (16), we obtain .
In the case of neglecting the connections between the two horizons, the black hole horizon and the cosmological horizon are two independent thermodynamic systems. Due to different horizon temperatures, the spacetime cannot be in thermal equilibrium. After considering the connections between the both horizons, from Eq.(17) one can see that only one effective temperature T ef f left.

IV. PHASE TRANSITION IN GAUSS-BONNET-DS BLACK HOLE SPACETIME
Recently, there are many works on the phase transition of black hole thermodynamic system. One can also compare the critical behaviors of van der Waals liquid-gas system and black hole system. Nevertheless, in dS spacetime there are the cosmological horizon and the black hole horizon. Generally, equilibrium cannot be achieved unless for extremal black hole. In this section, we employe the effective thermodynamic quantities to study the phase transition and critical behaviors of charged Gauss-Bonnet-dS black hole. This may avoid the apprehension of equilibrium of the black hole system.
Comparing with the Van der Waals equation Here, v = V /N is the specific volume of the fluid, P its pressure, T its temperature, and k is the Boltzmann constant.
Similarly, for the GB-dS black hole we can set the specific volume as According to Eq.(18), one can employ the two equations to calculate the critical temperature, critical pressure and the critical specific volume. We can ascertain the critical values (r 0 c , x 0 ) for given Q, n andα. According to these values, we can obtain the critical temperature, critical electric potential, critical pressure and critical volume. As is shown Table I, the critical temperature and critical pressure increase with the dimension of spacetime for fixed Q andα. However, they decrease with the increase of the Gauss-Bonnet parameterα for fixed spacetime dimension and electric charge Q. In order to describe the relation of P ef f and V in the vicinity of critical temperature, we plot the curves of P ef f -V at different temperatures. When the effective temperature T ef f > T c ef f , the stable condition ∂P ∂V T ef f < 0 can be satisfied. In the case of T ef f < T c ef f , there exist a portion in the curve where ∂P ∂V T ef f > 0, thus in these parts the system is unstable. So phase transition may occur only at T ef f = T c ef f . According to Ehrenfest classification, when the chemical potential and its first derivative are continuous, whereas the second derivative of chemical potential is discontinuous, this kind of phase transition is called the second-order phase transition. For Van der Waals equation, at the critical point no latent and sudden change in volume happen between liquid phase and gas phase. According to Ehrenfest classification, this kind of phase transition is continuous.
To understand the types of the phase transition for the GB-dS black hole, we should calculate the specific heat at constant pressure C P , the volume expansivity β, and the isothermal compressibility κ: We also depict the curves of C P − x, κ T − x and β − x in the Fig.2, Fig.3 and Fig.4 respectively. From these curves, we find that, for the charged GB-dS black hole, there exists an infinite peak in these curves. Moreover, in Fig.5, one can see that the curves of S − T ef f are continuous at the critical point. According to Ehrenfest, the phase transition of the charged GB-dS black hole should be the second-order one, which is similar to the RN-(A)dS black hole.

V. DISCUSSION AND CONCLUSION
In this paper, we adopted Ehrenfests classification to study the phase transition of the charged GB-dS black hole. After introducing the connection between the thermodynamic quantities corresponding to the black hole horizon and the cosmological horizon, we give the effective thermodynamic quantities, such as effective pressure P ef f , effective temperature T ef f , etc, of the black hole. From the equation of state described by the effective thermodynamic quantities, we study the P − V criticality. As illustrated in Fig.1, it exhibits a similar phase transition to Van der Waals equation.
By treating the effective cosmological constant as the thermodynamic pressure, in the extended phase space, we completely follow the Ehrenfest classification to explore the type of the phase transition of the charged GB-dS black hole. It is shown that when the entropy is a continuous function of temperature. However, the heat capacity C P , the isothermal compressibility κ T and the expansion coefficient β are all divergent at the critical point. This means this kind of phase transition for the charged GB-dS black hole is second order.