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$4D$ AdS Einstein-Gauss-Bonnet black hole with Yang-Mills field and its thermodynamics

We derive an exact black hole solution for the Einstein-Gauss-Bonnet gravity with Yang-Mills field in $4D$ AdS spacetime and investigate its thermodynamic properties to calculate exact expressions for the black hole mass, temperature, entropy and heat capacity. The thermodynamic quantities get modification in the presence of Yang-Mills field, however, entropy remains unaffected by the Yang-Mills charge. The solution exhibits $P-v$ criticality and belongs to the universality class of Van der Waals fluid. We study the effect of Gauss-Bonnet coupling and Yang-Mills charge on the critical behaviour and black hole phase transition. We observe that the values of critical exponents increase with the Yang-Mills charge and decrease with the Gauss-Bonnet coupling constant.