$P-V$ 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) 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.