Euclidean and Hamiltonian thermodynamics for regular black holes

We investigate the thermodynamic properties of the Hayward regular black hole using both Euclidean path integral and Hamiltonian methods, in asymptotically anti--de Sitter, Minkowski, and de Sitter spacetimes. With the inclusion of matter fields which act as a source for the regular black hole geometry, an effective temperature emerges that differs from the conventional definition related to the Killing surface gravity. We posit that this temperature is the appropriate choice for studying thermodynamic phenomena, by demonstrating consistency between the Euclidean and Hamiltonian formulations in the appropriate limits. We examine the thermodynamic properties and phase structure of the Hayward black hole in the canonical ensemble and show that, counter to some earlier indications, standard mean-field theory critical behavior is observed when the cosmological constant is treated as a thermodynamic pressure. We note the absence of a Hawking-Page transition, and conjecture that quantum gravity corrections which are suitably strong to regulate the Schwarzschild singularity generically prevent the transition from occurring. We also show that the Smarr relation remains linear in all cases, despite the absence of a linearity proof for nonlinear electrodynamic theories with nonsymmetry inheriting fields.