Refined thermodynamic analysis of perfect fluid dark matter black holes in a phantom background

In this study, we examine the thermodynamics of black holes immersed in perfect fluid dark matter (PFDM) by employing the Misner-Sharp energy framework. We extend the analysis to include the thermal fluctuations of these black holes when size reduces to small size, redefining key thermodynamic variables such as pressure, volume, temperature, internal energy, and entropy within the context of PFDM. Using these newly defined variables, we systematically calculate the enthalpy, Gibbs free energy, and specific heat of PFDM black holes, incorporating perturbative thermal corrections. To assess the stability of these black holes, we perform a detailed graphical analysis of the specific heat as a function of the horizon radius, providing insights into the thermal stability of PFDM black holes. This comprehensive approach enhances our understanding of the thermodynamic properties and stability of black holes in the presence of dark matter.