$P-V$ Criticality of Conformal Gravity Holography in Four Dimensions

{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) \emph{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 \emph{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 \emph{{the} critical ratio {has a constant value}} and {it is} independent of the {non-trivial} \emph{Rindler parameter ($a$)}. Finally, we derive {the} \emph{reduced equation of state} in terms of {the} \emph{reduced temperature}, {the} \emph{reduced volume} and {the} \emph{reduced pressure} respectively.
Comment: Accepted in MPLA, Modern Physics Letters A, 2018