First law and black hole thermodynamics of black hole in Dehnen (1,4,12) dark matter environment

Abstract

In this work, we present a comprehensive thermodynamic analysis of a static, spherically symmetric black hole embedded in a Dehnen $(1,4,\frac{1}{2})$ dark matter halo. By constructing an enthalpy-based mass function adapted to the Dehnen profile, we derive explicit expressions for the Hawking temperature, the Gibbs free energies and the specific heat capacities. Building upon these results, we explore the thermodynamic phase structure using an implicit approach in both the $G-T$ and $c_H-T$ frameworks. This work uncovers Van der Waals-like behavior and the corresponding critical points, providing a coherent picture of the black hole + dark matter stability through the behavior of Gibbs free energy and the heat capacity. Comparison to the bearly Schwarzschild black hole shows that the presence of the Dehnen $(1,4,\frac{1}{2})$ type dark matter halo introduces substantial modifications to the thermodynamic behavior of the black hole. In particular, the halo plays a decisive role in regulating of black hole phase transitions.