Motivated by recent developments in black hole thermodynamics, we investigate van der Waals phase transitions of charged black holes in massive gravity. We find that massive gravity theories can exhibit strikingly different thermodynamic behavior compared to that of Einstein gravity, and that the mass of the graviton can generate a range of new phase transitions for topological black holes that are otherwise forbidden.

In this paper, we study massive gravity in the presence of Born-Infeld nonlinear electrodynamics. First, we obtain metric function related to this gravity and investigate the geometry of the solutions and find that there is an essential singularity at the origin (r = 0). It will be shown that due to contribution of the massive part, the number, type and place of horizons may be changed. Next, we calculate the conserved and thermodynamic quantities and check the validation of the first law of thermodynamics. We also investigate thermal stability of these black holes in context of canonical ensemble. It will be shown that number, type and place of phase transition points are functions of different parameters which lead to dependency of stability conditions to these parameters. Also, it will be shown how the behavior of temperature is modified due to extension of massive gravity and strong nonlinearity parameter. Next, critical behavior of the system in extended phase space by considering cosmological constant as pressure is investigated. A study regarding neutral Einstein-massive gravity in context of extended phase space is done. Geometrical approach is employed to study the thermodynamical behavior of the system in context of heat capacity and extended phase space. It will be shown that GTs, heat capacity and extended phase space have consistent results. Finally, critical behavior of the system is investigated through use of another method. It will be pointed out that the results of this method is in agreement with other methods and follow the concepts of ordinary thermodynamics.

In this paper, we consider linearly charged dilatonic black holes and study their thermodynamical behavior in the context of phase transitions and thermodynamic geometry. We show that, depending on the values of the parameters, these black holes can undergo two types of phase transition. We also find that there are three critical behaviors near the critical points for these black holes: nonphysical unstable to physical stable, large to small, and small to large black-hole phase transitions. Next, we employ a thermodynamical metric to study the thermodynamical geometry of these black holes. We show that the characteristic behavior of the Ricci scalar of this metric enables one to recognize the type of phase transition and critical behavior of the black holes near phase-transition points. Finally, we will extend thermodynamical space by considering the dilaton parameter as an extensive parameter. With this consideration, we will show that the Weinhold, Ruppeiner, and Quevedo metrics provide extra divergencies that are not related to any phase-transition point, whereas our new method provides an effective mechanism to study phase transitions.

Motivated by high interest in the close relation between string theory and black hole solutions, in this paper, we take into account the Einstein-Gauss-Bonnet Lagrangian in the context of massive gravity. We examine the possibility of black hole in this regard, and discuss the types of horizons. Next, we calculate conserved and thermodynamic quantities and check the validity of the first law of thermodynamics. In addition, we investigate the stability of these black holes in context of canonical ensemble. We show that number, type and place of phase transition points may be significantly affected by different parameters. Next, by considering cosmological constant as thermodynamical pressure, we will extend phase space and calculate critical values. Then, we construct thermodynamical spacetime by considering mass as thermodynamical potential. We study geometrical thermodynamics of these black holes in context of heat capacity and extended phase space. We show that studying heat capacity, geometrical thermodynamics and critical behavior in extended phase space lead to consistent results. Finally, we will employ a new method for obtaining critical values and show that the results of this method are consistent with those of other methods.