In this paper, we investigate the thermodynamic properties of rotational Kiselev black holes (KBH). Specifically, we use the first-order approximation of the event horizon (EH) to calculate thermodynamic properties for general equations of state [Formula: see text]. These thermodynamic properties include areas, entropies, horizon radii, surface gravities, surface temperatures, Komar energies and irreducible masses at the Cauchy horizon (CH) and EH. We study the products of these thermodynamic quantities, we find that these products are determined by the equation of state [Formula: see text] and strength parameter [Formula: see text]. In the case of the quintessence matter [Formula: see text], radiation [Formula: see text] and dust [Formula: see text], we discuss their properties in detail. We also generalize the Smarr mass formula and Christodoulou–Ruffini mass formula to rotational KBH. Finally, we study the phase transition and thermodynamic geometry for rotational KBH with radiation [Formula: see text]. Through analysis, we find that this phase transition is a second-order phase transition. Furthermore, we also obtain the scalar curvature in the thermodynamic geometry framework, indicating that the radiation matter may change the phase transition condition and properties for Kerr black hole.
Holographic insulator/superconductor phase transition via matching method and thermodynamic geometry approach
