Topological thermodynamics of generalized Bardeen black hole

Abstract

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 \(w=+1\) and \(w=-1\), resulting in a total topological number of \(W=0\). Within the more recent universal thermodynamic-topology classification, this stable-to-unstable defect ordering corresponds to the previously established \(W^{0+}\) class, whereas the Schwarzschild limit is consistent with the \(W^{1-}\) class. The parameters \(\alpha \) and \(\beta \) 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 \(W=-1\). 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.