Holographic Extended Thermodynamics of deformed AdS-Schwarzschild black hole

Abstract: We investigate the thermodynamics and phase structure of a deformed AdS-Schwarzschild black hole, generated via the gravitational decoupling (GD) method as a minimal geometric deformation. In the bulk canonical ensemble, our results exhibit a van der Waals-type first-order phase transition in addition to the Hawking-Page transition, in the suitable parameter regime. Further, we compute the critical exponents characterising the bulk transition, confirming their consistency with mean-field theory predictions. Exploiting the exact holographic dictionary between extended black hole thermodynamics and the dual conformal field theory (CFT), we extend this analysis to the boundary and uncover a rich array of phase transitions and critical phenomena across three distinct thermodynamic ensembles. In particular, for an ensemble of fixed volume and central charge, the dual CFT displays a van der Waals-like phase structure. Throughout, we emphasise the pivotal influence of the GD deformation parameter on the thermodynamic behaviour, and we elucidate its role in the confinement-deconfinement transitions characteristic of the deformed AdS-Schwarzschild geometry.