Thermal Analysis, Joule-Thomson Expansion and Hawking Sparsity of Mod(A)Max-AdS Black Hole Immersed in a Cloud of Strings

We investigate the thermodynamic behavior of a spherically symmetric Anti-de Sitter black hole in Mod(A)Max electrodynamics surrounded by a cloud of strings. Within the extended phase-space framework, we treat the cosmological constant as a pressure and interpret the black-hole mass as enthalpy, which enables a unified discussion of local stability, global phase structure, and Joule–Thomson expansion. We analyze the Hawking temperature, Gibbs free energy, and heat capacity, and show how the string-cloud parameter, the Mod(A)Max deformation, and the electric charge reshape the physical domain, the stability windows, and the small/large black-hole transition pattern. We further characterize the critical behavior and demonstrate that a van der Waals–like phase structure arises only in the physical sector, while the alternate branch does not admit a genuine critical point. For the Joule-Thomson process, we determine the inversion curve and the corresponding isenthalpic trajectories, highlighting how the model parameters control the cooling/heating regimes and can generate terminating isenthalpic behavior at sufficiently large charge. Finally, we examine the sparsity of Hawking radiation and discuss how the underlying parameters influence the temporal discreteness of the emitted flux, particularly near extremality and in the large-radius AdS regime.