Probing the Thermodynamic Phase Structure of Black Holes through Euler characteristic

In this work, we attempt to explore a possible connection between thermodynamic topology and the thermodynamic geometry formulation of black hole thermodynamics. We study the topological structure of black hole thermodynamic phase spaces by calculating the Euler characteristic (EC) using four well-known thermodynamic geometries: Weinhold, Ruppeiner, Geometrothermodynamics (GTD), and HPEM. We interpret the Euler characteristic as an indicator of the degree of microscopic interactions within the thermodynamic system. As the system approaches the spinoidal curve, the interaction strength increases significantly, eventually driving a phase transition. Beyond the spinoidal region, the interactions begin to weaken, and the system gradually stabilizes into a new phase configuration, reflected by a corresponding change in the topological structure of the thermodynamic state space.