Following previous study about AdS-Schwarzschild black holes minimally coupled to a cloud of strings in the context of massive gravity [1] and inspired by strong connection between Gauss-Bonnet Gravity and heterotic string theory, in this paper, we first take into account the Gauss-Bonnet term and we study thermodynamics and critical behavior of these black holes in the extended phase space. The effects of Gauss-Bonnet, massive, and string cloud parameters on the criticality of these black holes has been investigated. It can be seen that the Gauss-Bonnet and massive parameters have opposite effects on the criticality and phase transition of the solutions. We also observe that the increase in the value of the string cloud parameter above a critical value, eliminates the van der Waals like behavior of these solutions. Also, the Joule-Thomson effect is not observed. Then we examine thermal stability of these black holes in canonical ensemble by calculating the heat capacity. In addition, we explore critical behavior in extended phase space by employing heat capacity and consequently, we observe that the results are in agreement with the previous results from the usual method in section 3.
The Einstein AdS black brane with a cloud of strings background in context of massive gravity is introduced. There is a momentum dissipation on the boundary because of graviton mass on the bulk. The ratio of shear viscosity to entropy density is calculated for this solution. This value violates the KSS bound if we apply the Dirichlet boundary and regularity on the horizon conditions. Our result shows that this value is independent of the cloud of strings.
In this paper, we study AdS-Schwarzschild black holes in four and five dimensions in dRGT minimally coupled to a cloud of strings. It is observed that the entropy of the string cloud and massive terms does not affect the black hole entropy. The observations about four dimensions indicate that the massive term in the presence of external string cloud cannot exhibit Van der Waals-like behavior for AdS-Schwarzschild black holes and, therefore there is only the Hawking–Page phase transition. In contrast, in five dimensions, the graviton mass modifies this behavior through the third massive term, so that a critical behavior and second-order phase transition is deduced. Also, the Joule–Thomson effect is not observed. The black hole stability conditions are also studied in four and five dimensions and a critical value for the string cloud parameter is presented. In five dimensions a degeneracy between states for extremal black holes is investigated. After studying black holes as thermodynamic systems, we consider such systems as heat engines, and finally the efficiency of them is calculated.
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