How a minimal length scale modifies thermodynamics of RN AdS Black Holes?

Abstract

We investigate the thermodynamic modifications of the Reissner-Nordstroem anti-de Sitter (RN AdS) black hole induced by a minimal length scale,which naturally emerges in $κ$-deformed space-time. By constructing the modified metric via phase-space commutation relations,we derive the deformed Hawking temperature, entropy, and enthalpy. We analyze the thermal stability through the heat capacity and study the P-V criticality, revealing that the black hole undergoes a small-to-large phase transition analogous to the Van der Waals system, albeit with a critical ratio slightly lowered by non-commutativity. Furthermore, we examine the Joule-Thomson expansion and find that the non-commutative (NC) parameter expands the cooling region in the temperature-pressure plane. Our results demonstrate that while the overall thermodynamic analogy with the Van der Waals fluid persists, the minimal length scale systematically deform the coexistence region and inversion curves, offering potential observational signatures for quantum gravity.