From Nonextremal to Extremal: Entropy of Reissner-Nordström and Kerr black holes Revisited

In this paper, we derive the entropy of Reissner-Nordström (RN) and Kerr black holes using the Hawking–Gibbons path integral method. We determine the periodicity of the Euclidean time coordinate using two approaches: first, by analyzing the near-horizon geometry, and second, by applying the Chern–Gauss–Bonnet (CGB) theorem. For non-extremal cases, both these methods yield a consistent and unique periodicity, which in turn leads to a well-defined expression for the entropy. In contrast, the extremal case exhibits a crucial difference. The absence of a conical structure in the near-horizon geometry implies that the periodicity of the Euclidean time is no longer uniquely fixed within the Hawking–Gibbons framework. The CGB theorem also fails to constrain the periodicity, as the corresponding Euler characteristic vanishes. As a result, the entropy cannot be uniquely determined using either method.