Thermodynamic properties and geometries of bardeen black hole surrounded by string clouds

In this work, we investigate the thermodynamic properties of Bardeen black hole which is coupled with cloud of strings and minimally coupled to nonlinear electrodynamics. The modified entropy in the form of Sharma-Mittal entropy is used to discuss these properties which include mass, temperature, pressure, Gibbs free energy and trace of Hessian matrix. We obtain stable behavior along with physical solution for some specific values of parameters a and q. Furthermore, our work provides a thermodynamic metric using the Hessian matrix of black hole mass, changing the conformal connection between Quevedo and Ruppeiner's geometries. Investigating the first principle of thermodynamics for regular black holes, such as the Bardeen AdS black hole, exposes significant behavior changes during phase transitions in an extended phase space.