Thermodynamic Properties of a Pseudo-Isothermal Black Hole and Photon Sphere

This paper explores the thermodynamic geometry of a non-rotating, spherically symmetric black hole embedded in a pseudo-isothermal dark matter (DM) halo. The study carefully analyzes the metric function, horizon structure, mass function, surface gravity, Hawking temperature, Helmholtz and Gibbs free energies, and the nontrivial behavior of the black hole’s heat capacity. We find that the enhanced halo influence in the vicinity of the black hole suppresses Hawking radiation and may hinder thermodynamic phase transitions. Our results further indicate that the mass function is positively correlated with entropy, revealing a direct thermodynamic coupling between the distribution of gravitational mass and the global thermodynamic state of the system. Overall, the inclusion of a pseudo-isothermal dark matter halo alters the thermodynamic properties of the black hole and induces stability features that are absent in vacuum configurations. Finally, we show that increasing both the central core density and the radial extent of the DM halo decreases the photon-sphere radius and the shadow size, highlighting the pronounced role of dark matter environments in shaping black hole observables and thereby improving both theoretical understanding and prospects for observational tests. In this work, particular attention is devoted to the halo parameters, allowing a thorough assessment of how they influence the thermodynamic behavior of the black hole, as well as the properties of the photon sphere and shadow.