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.

We show by explicit computations that there is a superficial inconsistency between the conventional first law of black hole thermodynamics and Bekenstein-Hawking area law for three types of regular black holes. The corrected form of the first law for these regular black holes is given. The derivation relies on the general structure of the energy-momentum tensor of the matter fields. When the black hole mass parameter M is included in the energymomentum tensor, the conventional form of the first law should be modified with an extra factor. In this case, the black hole mass M can no longer be considered as the internal energy of the regular black holes.

We obtain a magnetically charged regular black hole in general relativity. The source to the Einstein field equations is nonlinear electrodynamic field in a physically reasonable model of nonlinear electrodynamics (NED). "Physically" here means the NED model is constructed on the basis of three conditions: the Maxwell asymptotic in the weak electromagnetic field limit; the presence of vacuum birefringence phenomenon; and satisfying the weak energy condition (WEC). In addition, we analyze the thermodynamic properties of the regular black hole in two ways. According to the usual black hole thermodynamics, we calculate the heat capacity at constant charge, from which we know the smaller black hole is more stable. We also employ the horizon thermodynamics to discuss the thermodynamic quantities, especially the heat capacity at constant pressure. PACS numbers: 04.70.Dy I. INTRODUCTION The first well-known model of nonlinear electrodynamics(NED) is the Born-Infeld theory(BI), which is proposed to obtain a finite electron self-energy[1]. Heisenberg and Euler found that due to the presence of virtual charged particles the one-loop quantum correction in quantum electrodynamics will give nonlinear contribution[2]. Not only that, the virtual particles will result in " polarization of the vacuum ". In this case, the vacuum behaves like a polarizable continuum and should exhibit the phenomenon of birefringence[3-5]. These effects can be observed in experiments such as PVLAS[6] and BMV[7] and the experimental results can put some restrictions on the parameters introduced in the NED models. Vacuum birefringence is a nonlinear effect. Therefore, we expect that in a physically reasonable model of NED, there should exist the effect of vacuum birefringence. In this sense, the usual BI theory is not a physically allowable NED model because of the absence of vacuum birefringence. However, in the generalized BI model with two parameters the vacuum birefringence may exist[8]. The nonlinear effects in electrodynamics are significant only *