Regular black holes and the first law of black hole mechanics

Singularity-free regular black holes are a popular alternative to the singular mathematical black holes predicted by general relativity. Here, we derive a generic condition that spherically symmetric dynamical regular black holes must satisfy to be compatible with the first law of black hole mechanics based on an expression for the surface gravity at the outer horizon. We examine the dynamical generalizations of models typically considered in the literature and demonstrate that none of them satisfies the condition required for compatibility with the first law, suggesting that modifications are required to maintain its physical meaning. We show that the need for corrections is inherently linked to the introduction of a minimal length scale and can therefore be seen as a direct consequence of the spacetime regularization. We explicitly identify the additional work terms in the extended first law, comment on their thermodynamic interpretation, and show that the linear coefficient of the Misner-Sharp mass suffices to determine the relevant thermodynamic properties.