In this paper, we study the phase structure and equilibrium state space geometry of charged topological dilaton black holes in (n + 1)-dimensional anti-de Sitter spacetime. By considering the pairs of parameters (P ∼ V ) and (Q ∼ U ) as variables, we analyze the phase structure and critical phenomena of black holes and discuss the relation between the two kinds of critical phenomena. We find that the phase structures and critical phenomena drastically depend on the cosmological constant l (or the static electric charge Q of the black holes), dimensionality n and dilaton field Φ.

It is well known that there are the black hole horizon and the cosmological horizon for the ReissnerNordström-de Sitter (RN-dS) spacetime. The thermodynamic quantities on the both horizons satisfy the first law of the black hole thermodynamics, respectively; moreover, there are some additional connections between them. In this paper by considering the relations between the two horizons we give the effective thermodynamic quantities in (n + 2)-dimensional RN-dS spacetime. The thermodynamic properties of these effective quantities are analyzed; moreover, the critical temperature, critical pressure, and critical volume are obtained. We carry out an analytical check of the Ehrenfest equations and prove that both Ehrenfest equations are satisfied. So the spacetime undergoes a second-order phase transition at the critical point. This result is consistent with the nature of a liquid-gas phase transition at the critical point, hence deepening the understanding of the analogy of charged dS spacetime and liquid-gas systems.

After introducing the connection between the black hole horizon and the cosmological horizon, we discuss the thermodynamic properties of Reissner-Nordstrom-de Sitter (RN-dS) spacetime. We present the condition under which RN-dS black hole can exist. Employing Ehrenfest' classification we conclude that the phase transition of RN-dS black hole is the second-order one. The position of the phase transition point is irrelevant to the electric charge of the system. It only depends on the ratio of the black hole horizon and the cosmological horizon.keywords:Reissner-Nordstrom-de Sitter black hole, critical phenomena, the second order phase transition Contents