First law for nonsingular black holes in 2D dilaton gravity

A central issue in the thermodynamics of nonsingular black holes is the apparent violation of the first law. In this work, we use 2D dilaton gravity as a simple theoretical setting to study this issue. We systematically construct a broad class of nonsingular black hole solutions with metric function A(x)=f(x)+c, through a procedure that is considerably simpler than in higher-dimensional theories. Using the Iyer-Wald covariant phase space formalism, we derive the correct energy formula and establish a consistent first law for this entire class of solutions. The apparent violation of the first law in a previous work arises because the energy used therein is not the Hamiltonian conjugate to the fixed time-translation generator adopted. Under the boundary conditions and normalization convention specified in the present work, the Iyer-Wald Hamiltonian energy is E=−c2, and the first law is restored. Moreover, the energy formula agrees with the Casimir function in 2D dilaton gravity, thus confirming its interpretation as the physical black hole energy. Our results clarify the correct first law for 2D nonsingular black holes and may provide insights into the first law of nonsingular black holes in higher dimensions.