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In this work, we investigate the thermodynamic properties of rotationalKiselev black holes (KBH). Specifically, we use the first-order approximationof the event horizon (EH) to calculate thermodynamic properties for generalequations of state $\omega$. These thermodynamic properties include areas,entropies, horizon radii, surface gravities, surface temperatures, Komarenergies and irreducible masses at the Cauchy horizon (CH) and EH. We study theproducts of these thermodynamic quantities, we find that these products aredetermined by the equation of state $\omega$ and strength parameter $\alpha$.In the case of the quintessence matter ($\omega=-2/3$), radiation($\omega=1/3$) and dust ($\omega=0$), we discuss their properties in detail. Wealso generalize the Smarr mass formula and Christodoulou-Ruffini mass formulato rotational KBH. Finally we study the phase transition and thermodynamicgeometry for rotational KBH with radiation ($\omega=1/3$). Through analysis, wefind that this phase transition is a second order phase transition.Furthermore, we also obtain the scalar curvature in the thermodynamic geometryframework, indicating that the radiation matter may change the phase transitioncondition and properties for Kerr black hole.

Thermodynamics and phase transition in rotational Kiselev black hole