Thermodynamics and phase transitions of $\kappa$-deformed Schwarzschild-AdS black holes

Abstract

We investigate the thermodynamics of the Schwarzschild-AdS black hole in the framework of \(\kappa\)-deformed non-commutative geometry by constructing an effective \(\kappa\)-deformed Schwarzschild-AdS metric from the \(\kappa\)-deformed Newtonian potential. In the extended phase space, we derive a modified first law and the corresponding Smarr relation by treating the \(\kappa\)-deformation parameter as an additional thermodynamic variable and identifying its conjugate potential. Our analysis shows that \(\kappa\)-deformation induces critical behaviour and phase transitions in an uncharged Schwarzschild-AdS black hole, with a critical ratio \(P_c v_c/T_c \simeq 0.370\) that is independent of the deformation parameter and close to the Van der Waals value. Interestingly, the \(G - T\) curve exhibits a peculiar double-loop structure, deviating from the standard swallow-tail behaviour associated with a first-order phase transition.