Thermodynamics and $P$-$V$ Criticality of Charged AdS Black Holes with a Cloud of Strings in Kalb-Ramond Gravity

Abstract: We investigate the extended phase-space thermodynamics and $P$--$V$ criticality of electrically charged anti-de Sitter (AdS) black holes in Kalb--Ramond bumblebee gravity in the presence of a spherically symmetric cloud of strings. The background Kalb--Ramond field induces Lorentz symmetry violation through a dimensionless parameter $\ell$, while the string cloud is characterized by a parameter $α$, both entering the lapse function and deforming the geometry. Interpreting the ADM mass as enthalpy, we derive the main thermodynamic quantities, Hawking temperature, entropy, thermodynamic volume, Gibbs free energy, internal energy, and specific heat, and analyze how $(\ell,α)$ jointly affect stability and phase structure. We obtain a Van der Waals--type equation of state, compute the critical point $(P_c,T_c,v_c)$, and show that, although the critical scales depend nontrivially on $\ell$ and $α$, the universal ratio $P_c v_c/T_c = 3/8$ is preserved. Using the thermodynamic topology approach, we determine that the total topological charge of the black hole solution remains $W=1$, placing it in the same topological class as the Reissner--Nordström--AdS black hole. Finally, by studying the Joule--Thomson expansion, we derive the inversion curve and find that the minimal inversion temperature satisfies $T_i^{\rm min}/T_c = 1/2$, as in the Reissner--Nordström--AdS case, indicating that Lorentz violation and the string cloud deform the thermodynamic scales without changing the underlying universality class.