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

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 $P_c{v}_c\mathrm{/}{T}_c\simeq 0.370$ 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.