Holographic extended thermodynamics of deformed AdS-Schwarzschild black hole

We investigate the thermodynamics and phase structure of the deformed AdS-Schwarzschild black hole, generated via the gravitational decoupling (GD) method. 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, in the fixed $(\mathcal{V},C)$ ensemble, the dual CFT exhibits a Hawking-Page-type transition. However, in the fixed $(p,C)$ ensemble, the deformation parameter leads to a distinct thermodynamic behaviour in which multiple branches become unstable, leaving a single thermodynamically stable phase, thus marking a clear departure from the standard van der Waals scenario. 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.