Extended thermodynamics and $P-v$ Criticality of Kalb-Ramond black hole coupled with nonlinear electrodynamics
We present an exact black hole solution in anti-de Sitter (AdS) spacetime with a Kalb-Ramond field coupled to nonlinear electrodynamics (NLED), characterized by mass, magnetic monopole charge, and Lorentz-violating parameters. The geometry admits two horizons (inner and outer) that coalesce into a degenerate horizon at a critical monopole charge. Beyond this critical point, no black hole solutions exist. In the limit of vanishing Lorentz-violating parameters, the solution reduces to the modified Kalb-Ramond and Bardeen black holes, while suitable parameter choices reproduce the Reissner-Nordström-AdS and Schwarzschild-AdS geometries. We analyze the thermodynamics of the solution by computing the Hawking temperature, entropy, specific heat, and Gibbs free energy. The NLED source introduces nontrivial modifications: the Hawking temperature displays nonmonotonic behavior with possible local extrema, the entropy deviates from the standard area law, and the specific heat may assume negative values, signaling thermodynamic instabilities. The Gibbs free energy exhibits swallow-tail structures, indicative of first-order phase transitions. Furthermore, we derive the first law of black hole thermodynamics in the extended phase space, together with the Smarr relation, and confirm their validity for the Kalb-Ramond black holes with NLED sources. Our findings highlight the rich thermodynamic structure induced by Lorentz-violating effects and nonlinear electrodynamics in AdS black hole backgrounds.