Parthapratim Pradhan

We study the extended phase space thermodynamics of black holes in massive gravity. Particularly, we examine the critical behaviour of this black hole using the extended phase space formalism. Extended phase space in a sense that in which the cosmological constant should be treated as a thermodynamic pressure and its conjugate variable as a thermodynamic volume. In this phase space, we derive the black hole equation of state, the critical pressure, the critical volume and the critical temperature at the critical point. We also derive the critical ratio of this black hole. Moreover, we derive the black hole reduced equation of state in terms of the reduced pressure, the reduced volume and the reduced temperature. Furthermore, we examine the Ehrenfest equations of black holes in massive gravity in the extended phase space at the critical point. We show that the Ehrenfest equations are satisfied of this black hole and the black hole encounters a second order phase transition at the critical point in the said phase space. This is re-examined by evaluating the Pregogine-Defay ratio ($\Pi$). We determine the value of this ratio is $\Pi =1$. The outcome of this study is completely analogous to the nature of liquid-gas phase transition at the critical point. This investigation also further gives us the profound understanding between the black hole of massive gravity with the liquid-gas system.

{We examine the critical behaviour i. e. $P-V$ criticality of conformal gravity (CG) in an extended phase space in which the cosmological constant should be interpreted as a thermodynamic pressure and the corresponding conjugate quantity as a thermodynamic volume.} The main potential point of interest in CG is that there exists a non-trivial \emph{Rindler parameter ($a$)} in the {spacetime geometry. This geometric parameter has an important role to construct a model for gravity at large distances where the parameter "$a$" actually originates}. We also investigate the effect of the said parameter on the black hole (BH) thermodynamic equation of state, critical constants, Reverse Isoperimetric Inequality, first law of thermodynamics, Hawking-Page phase transition and Gibbs free energy for this BH. We speculate that due to the presence of the said parameter, there has been a deformation in the shape of the isotherms in the $P-V$ diagram in comparison with the charged-AdS (anti de-Sitter) BH and the chargeless-AdS BH. Interestingly, we find that the critical ratio for this BH is ${\rho }_c=\frac{P_c{v}_c}{T_c}=\frac{\sqrt{3}}{2}\left(3\sqrt{2}-2\sqrt{3}\right)$, which is greater than the charged AdS BH and Schwarzschild-AdS BH i.e. ${\rho }_c^{CG}:{\rho }_c^{Sch-AdS}:{\rho }_c^{RN-AdS}=0.67:0.50:0.37$. The symbols are defined in the main work. Moreover, we observe that {the critical ratio has a constant value} and it is independent of the non-trivial \emph{Rindler parameter ($a$)}. Finally, we derive the reduced equation of state in terms of the reduced temperature, the reduced volume and the reduced pressure respectively.