Thermodynamic analysis of charged anti-de Sitter black holes with cloud of strings in Einstein-bumblebee gravity via Bekenstein–Hawking and Tsallis entropies

Abstract We investigate the thermodynamic properties of charged anti-de Sitter (AdS) black holes surrounded by a cloud of strings in bumblebee gravity. In this framework, the cloud-of-strings parameter α and the Lorentz-violating parameter ℓ modify the horizon structure, the Hawking temperature, the free energies, the specific heat and the critical behaviour in the extended phase–space description. We derive the corresponding equation of state and show that the system exhibits a small-large black hole phase transition of Van der Waals type. In particular, the critical quantities are deformed by both the cloud of strings and the bumblebee background, while the universal ratio is explicitly altered by Lorentz symmetry breaking. We also examine the Joule–Thomson expansion and analyse the associated inversion and isenthalpic curves, showing how the deformation parameters shift the boundary between heating and cooling regions. In addition, we extend the thermodynamic analysis to a Tsallis entropy-based framework and show that the non-extensive parameter δ significantly changes the temperature profile, stability windows, critical volume, free energies and sparsity of Hawking radiation. Our results reveal that the combined effects of the string cloud, Lorentz violation and non-extensive entropy lead to a substantially richer thermodynamic structure than that of the standard Reissner–Nordström–AdS black hole.