On the Validity of the Background Subtraction Method for Black Hole Thermodynamics in Matter-Coupled Gravity Theories

Abstract: The background subtraction method has long served as a practical tool for computing the Euclidean action and thermodynamic quantities of black holes. While its equivalence to the Iyer-Wald formalism is well understood in pure gravity theories, its validity in matter-coupled theories remains less clear and has even been questioned in the literature. In this work, we revisit this issue and demonstrate that the equivalence between the Euclidean action method and the Iyer-Wald formalism persists in matter-coupled scenarios. We apply the resulting formulation to two representative examples of such theories, and in both cases, the Euclidean approach performs smoothly. We further identify situations where the method may encounter subtleties due to the special properties of certain matter fields. Our results clarify when background subtraction remains reliable beyond pure gravity and when additional care is necessary.