Adiabatic evolution of Hayward black hole

Abstract: In this letter we use the Carath\'{e}odory's approach to thermodynamics, to construct the thermodynamic manifold of the Hayward black hole. The Pfaffian form representing the infinitesimal heat exchange reversibly is considered to be $\delta Q_{\mathrm{rev}}\equiv {\rm d}r_s-\mathcal{F}_H {\rm d}l$, previously obtained by Molina & Villanueva \cite{fmv20}, where $r_s$ is the Schwarzschild radius, $l$ is the Hayward's parameter responsible for the possible regularization of the Schwarzschild black hole, and $\mathcal{F}_H$ is the intensive variable called the Hayward's force. By solving the associated Cauchy problem, the adiabatic paths are confined to the non-extremal manifold, and therefore, the status of the second and third laws are preserved. Consequently, the extremal sub-manifold corresponds to the {adiabatically disconnected} boundary of the manifold. In addition, the merger of two extremal Hayward black holes is analyzed.