Black Hole Thermodynamics in Harada's inspired theory of gravity: Stability, Phase Structure and Geometrothermodynamics

In this paper, we investigate the thermodynamic properties of spherically symmetric, static black hole solutions within the framework of Conformal Killing Gravity (CKG). This is a modified theory of gravity that retains all solutions of General Relativity, while addressing some of its theoretical shortcomings and enriching gravitational phenomenology at large distances. We derive key thermodynamic quantities, including mass, temperature, heat capacity and Gibbs free energy, to examine the stability and phase structure of extended Schwarzschild-AdS and charged AdS black holes. Furthermore, employing the formalism of geometrothermodynamics, we analyze the behavior of the thermodynamic curvature scalar to identify critical points and characterize phase transitions. Our results demonstrate that the parameter \( \lambda \), which quantifies deviations from Einstein's theory, plays a pivotal role in shaping the thermodynamic behavior, resulting in new stability conditions and distinct phase transition patterns compared to those predicted by standard General Relativity.