Are regular black holes from pure gravity classified within the same thermodynamical topology?

Regular black holes, which avoid the essential central singularities, can be constructed through various methods, including nonlinear electrodynamics and quantum corrections. Recently, it was shown that via an infinite tower of higher-curvature corrections, one can obtain different regular black hole solutions in any spacetime dimension $D\ge 5$. Utilizing the concept of thermodynamical topology, we examine these black holes as topological thermodynamic defects, classifying them into distinct topological categories based on their generalized free energy. We demonstrate that the Hawking temperature of the black hole has at least one zero point at the small horizon radius limit. Under this fact, the regular black holes generated through the purely gravitational theories exhibit universal thermodynamical behaviors, strongly suggesting they belong to the same topological class. We present a comprehensive analysis of these properties, providing a clearer understanding of the fundamental nature of this type of regular black holes and their classification within the framework of thermodynamical topology.