Effect of a cloud of strings and quintessence on a phase transition of charged rotating AdS black holes

We investigate the phase portrait and the critical behavior in the extended phase space of one type of black hole solutions, the Kerr–Newman anti-de Sitter black hole in the presence of a spherically symmetric cloud of strings and quintessence. Regarding the cosmological constant as thermodynamic pressure, we give the essential steps to derive the expressions of thermodynamic quantities such as the Hawking temperature $T$, the Gibbs free energy $G$, and the heat capacity $C$. For this, we introduce an appropriate analytic approximation for the rotation parameter $a$ in terms of all important parameters: the cosmological constant, the entropy $S$, the parameters of quintessence and a cloud of strings, the charge $Q$, and the angular momentum $J$ of the system. The critical points are found numerically, and the $\widetilde T$–$\widetilde S$, $\widetilde G$–$\widetilde T$, and $\widetilde C$–$\widetilde S$ diagrams are obtained in the reduced parameter space. It is shown that these diagrams are dependent on the parameters of quintessence and the cloud of strings. The analysis of these diagrams leads to the conclusion that when a cloud of strings and quintessence are surrounding a charged rotating AdS black hole, these two extra sources of energy exert no effect on the existence of the small–large black-hole phase transition. Therefore, black holes exhibit a phase transition similar to the one in a van der Waals fluid. The approach adopted here has some important consequences. On the one hand, it can be used to analyze the critical behavior of the Kerr–Newman anti-de Sitter black hole with a cloud of strings and quintessence present in spacetime of a higher dimension (larger than $4$d), and on the other hand, it may be important in investigating the phase transition for other complicated black holes.