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Thermodynamics and phase transition in rotational Kiselev black hole

In this work, 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 $ω$. 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 $ω$ and strength parameter $α$. In the case of the quintessence matter ($ω=-2/3$), radiation ($ω=1/3$) and dust ($ω=0$), 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 ($ω=1/3$). 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.