Topological thermodynamics of generalized Bardeen black hole

Neves and Saa introduced a two-parameter spacetime that encompasses the Bardeen and Hayward geometries as special cases, together with a case whose f ( r ) function has the Simpson–Visser form. In this work, we employ the off-shell generalized Helmholtz free energy method to investigate the thermodynamic properties of this family within a topological framework. We begin by constructing the associated vector field and analyzing its zeros – specifically their winding numbers – to classify the thermodynamic branches and identify critical points associated with changes in global stability. Within the thermodynamic setup adopted here, the Bardeen, Hayward, and Simpson–Visser-type cases do not define independent global topological classes: whenever the off-shell conditions allow for two regular zeros, their winding numbers are $$w=+1$$ w = + 1 and $$w=-1$$ w = - 1 , resulting in a total topological number of $$W=0$$ W = 0 . Within the more recent universal thermodynamic-topology classification, this stable-to-unstable defect ordering corresponds to the previously established W⁰⁺ W 0 + class, whereas the Schwarzschild limit is consistent with the W¹⁻ W 1 - class. The parameters α α and β β are useful for determining the extremal bound, the critical radius, the inverse critical temperature, and the extent of the locally stable branch. In contrast, the Schwarzschild limit exhibits a monotonic curve for the inverse temperature and a single topological defect with $$W=-1$$ W = - 1 . These results provide a unified analytical characterization of how these configurations modify the local thermodynamic structure while preserving the global topological class of cases where a is finite.