Topological Thermodynamics of Generalized Bardeen Black Hole

Neves and Saa introduced a two parameter spacetime that includes the Hayward, Bardeen, and Simpson-Visser geometries as particular cases. In this work, we employ the generalized off-shell Helmholtz free energy method to investigate the thermodynamic properties of the generalized Bardeen black hole within a topological framework. We construct the associated vector field and analyze its zeros, whose winding numbers allow us to classify the thermodynamic branches and identify critical points associated with phase transitions. The regular black hole configurations exhibit two topological defects with opposite winding numbers, resulting in a vanishing total topological charge, while the Schwarzschild case contains a single unstable branch. Our results demonstrate how the regularization parameters affect the thermodynamic stability and phase structure of the spacetime.