Considering the cosmological constant Λ as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume as proposed in [1], we discuss the critical behavior of charged AdS black hole in arbitrary dimensions d. In particular, we present a comparative study in terms of the spacetime dimension d and the displacement of critical points controlling the transition between the small and the large black holes. Such behaviors vary nicely in terms of d. Among our result in this context consists in showing that the equation of state for a charged RN-AdS black hole predicts an universal number given by 2d−5 4d−8 . The three dimensional solution is also discussed.PACS numbers: 04.70.-s, 05.70.CeFew weeks after the submission of this work to Chinese Physics Letters (CPL) on August 15th, we learned that authors of [arxiv: 1208.6251] found similar results to ours, more notably the derivation of the same universal number given by 2d−5 4d−8 .

We investigate the heat properties of AdS Black Holes in higher dimensions. We consider the study of the corresponding thermodynamical properties including the heat capacity explored in the determination of the black hole stability. In particular, we compute the heat latent. To overcome the instability problem, the Maxwell construction, in the (T, S)-plane, is elaborated. This method is used to modify the the Hawking-Page phase structure by removing the negative heat capacity regions. Then, we discuss the thermodynamic cycle and the heat engines using the way based on the extraction of the work from a black hole solution.

Using an analytic analysis, we derive and solve the constraint equations obtained by folding the 2-spheres which are permuted under the outer automorphism groups of the toric graphs of affine ADE geometries in type II strings. We show that there are classes of solutions depending on the actions of on the elliptic curve E on which flat ADE bundles are fibred. We give two classes of four-dimensional N = 2 superconformal field theory solutions for affine BCFG geometries; one having the right space dimension but exhibiting mirror potentials with branch cuts and a second class sharing the main features with the solutions given by Mayr.

The principal focus of the present work concerns the critical behaviors of a class of three dimensional black holes with a scalar field hair. Since the cosmological constant is viewed as a thermodynamic pressure and its conjugate quantity as a volume, we examine such properties in terms of two parameters B and a. The latters are related to the scalar field and the angular momentum respectively. In particular, we give the equation of state predicting a critical universal number depending on the (B, a) moduli space. In the vanishing limit of the B parameter, we recover the usual perfect gas behavior appearing in the case of the non rotating BTZ black hole. We point out that in a generic region of the (B, a) moduli space, the model behaves like a Van der Waals system.