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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 behaviour 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 non-linear electrodynamic theories with non-symmetry
inheriting fields.
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