In this paper, we introduce the n-dimensional Lorentzian wormhole solutions of third order Lovelock gravity. In contrast to Einstein gravity and as in the case of Gauss-Bonnet gravity, we find that the wormhole throat radius, r 0 , has a lower limit that depends on the Lovelock coefficients, the dimensionality of the spacetime and the shape function. We study the conditions of having normal matter near the throat, and find that the matter near the throat can be normal for the region r 0 ≤ r ≤ r max , where r max depends on the Lovelock coefficients and the shape function. We also find that the third order Lovelock term with negative coupling constant enlarges the radius of the region of normal matter, and conclude that the higher order Lovelock terms with negative coupling constants enlarge the region of normal matter near the throat. * email address: mhd@shirazu.ac.ir

We explore the critical behavior of (n + 1)-dimensional topological Born-Infeld-dilaton black holes in an extended phase space. We treat the cosmological constant and the Born-Infeld (BI) parameter as the thermodynamic pressure and BI vacuum polarization which can vary. We obtain thermodynamic quantities of the system such as pressure, temperature, Gibbs free energy, and investigate the behaviour of these quantities. We also study the analogy of the van der Waals liquid-gas system with the Born-Infeld-dilaton black holes in canonical ensemble in which we can treat the black hole charge as a fixed external parameter. Moreover, we show that the critical values of pressure, temperature and volume are physical provided the coupling constant of dilaton gravity is less than one and the horizon is sphere. Finally, we calculate the critical exponents and show that although thermodynamic quantities depend on the dilaton coupling constant, BI parameter and the dimension of the spacetime, they are universal and are independent of metric parameters.

We investigate the critical behavior of an (n+1)-dimensional topological dilaton black holes, in an extended phase space in both canonical and grand-canonical ensembles, when the gauge field is in the form of power-Maxwell field. In order to do this we introduce for the first time the counterterms that remove the divergences of the action in dilaton gravity for the solutions with curved boundary. Using the counterterm method, we calculate the conserved quantities and the action and therefore Gibbs free energy in both the canonical and grand-canonical ensembles. We treat the cosmological constant as a thermodynamic pressure, and its conjugate quantity as a thermodynamic volume. In the presence of power-Maxwell field, we find an analogy between the topological dilaton black holes with van der Walls liquid-gas system in all dimensions provided the dilaton coupling constant α and the power parameter p are chosen properly. Interestingly enough, we observe that the power-Maxwell dilaton black holes admit the phase transition in both canonical and grand-canonical ensembles. This is in contrast to RN-AdS, Einstein-Maxwell-dilaton and Born-Infeld-dilaton black holes, which only admit the phase transition in the canonical ensemble. Besides, we calculate the critical quantities and show that they depend on α, n and p. Finally, we obtain the critical exponents in two ensembles and show that they are independent of the model parameters and have the same values as mean field theory.

In this paper, we take into account the dilaton black hole solutions of Einstein gravity in the presence of logarithmic and exponential forms of nonlinear electrodynamics. At first, we consider the cosmological constant and nonlinear parameter as thermodynamic quantities which can vary. We obtain thermodynamic quantities of the system such as pressure, temperature and Gibbs free energy in an extended phase space. We complete the analogy of the nonlinear dilaton black holes with Van der Waals liquid-gas system. We work in the canonical ensemble and hence we treat the charge of the black hole as an external fixed parameter. Moreover, we calculate the critical values of temperature, volume and pressure and show they depend on dilaton coupling constant as well as nonlinear parameter. We also investigate the critical exponents and find that they are universal and independent of the dilaton and nonlinear parameters, which is an expected result. Finally, we explore the phase transition of nonlinear dilaton black holes by studying the Gibbs free energy of the system. We find that in case of T > Tc, we have no phase transition. When T = Tc, the system admits a second order phase transition, while for T = T f < Tc the system experiences a first order transition. Interestingly, for T f < T < Tc we observe a zeroth order phase transition in the presence of dilaton field. This novel zeroth order phase transition is occurred due to a finite jump in Gibbs free energy which is generated by dilaton-electromagnetic coupling constant, α, for a certain range of pressure.

Recently, it was argued that charged Anti-de Sitter (AdS) black holes admit critical behavior, without extending phase space, similar to the Van der Waals fluid system in the Q 2 − Ψ plans where Ψ = 1/v (the conjugate of Q 2 ) is the inverse of the specific volume [1]. In this picture, the square of the charge of the black hole, Q 2 , is treated as a thermodynamic variable and the cosmological constant Λ is fixed. In this paper, we would like to examine whether this new approach toward critical behaviour of AdS black holes can work in other gravity such as Gauss-Bonnet (GB) gravity as well as in higher dimensional spacetime. We obtain the equation of state, Q 2 = Q 2 (Ψ, T ), Gibbs free energy and the critical quantities of the system, and study the effects of the GB couplingα on their behaviour. We find out that the critical quantities have reasonable values, provided the GB coupling constant,α, is taken small and the horizon topology is assumed to be (d−2)-sphere. Finally, we calculate the critical exponents and show that they are independent of the model parameters and have the same values as the Van der Waals system which is predicted by the mean field theory.