The Last Lost Charge And Phase Transition In Schwarzschild AdS Minimally Coupled to a Cloud of Strings

In this paper we study the Schwarzschild AdS black hole with a cloud of string background in an extended phase space and investigate a new phase transition related to the topological charge. By treating the topological charge as a new charge for black hole solution we study its thermodynamics in this new extended phase space. We treat by two approaches to study the phase transition behavior via both $T-S$ and $P-v$ criticality and we find the results confirm each other in a nice way. It is shown a cloud of strings affects the critical physical quantities and it could be observed an interesting Van der Waals-like phase transition in the extended thermodynamics. The swallow tail-like behavior is also observed in Free Energy-Temperature diagram. We observe in $a\to 0$ limit the small/large black hole phase transition reduces to the Hawking-Page phase transition as we expects. We can deduce that the impact of cloud of strings in Schwarzschild black hole can bring Van der Waals-like black hole phase transition.