Criticality and phase transition of Kerr–anti-de Sitter black hole with quintessence and cloud of strings

In this paper, a thermodynamical study of the Kerr–anti-de Sitter (KAdS) black hole (BH) encircled by quintessence matter and the cloud of strings is presented. We check the effects of the cloud parameter on the thermodynamical variables including Hawking temperature, entropy, Helmholtz free energy (HFE), Gibbs free energy (GFE), and specific heat capacity. These thermodynamic variables hold the Smarr–Gibbs–Dehum (SGD) relation in the appearance of quintessence and the cloud of the string parameter. Then we calculate the critical expressions of the developed variables in two different ways that are “Van-der-Waals (vdW)-type equation of state” (EoS) and the “Maxwell equal-area law” (MEAL). The latter technique is found to be more reasonable to examine the criticality of the complex BHs. Using MEAL, we plot the phase diagram in a temperature–entropy plane and obtain an isobar, which represents the co-existence region of liquid–gas phases. It is concluded that the uncharged BHs exhibit the same phase transition (PT) to vdW fluid as the temperature drops down its critical value. Also, we discuss the influence of thermal fluctuations (TF) on the stability of KAdS BH.