Thermodynamics and Phase Transition of Gravitational Global and Local Monopole

We compute the Hawking temperature of a regular self-gravitating ’t Hooft–Polyakov magnetic monopole in global and local monopole black holes. To this end, we apply two different methods: the tunneling method and the topological method, which both yield the standard Hawking temperatures of these two geometries. We then study a phase transition in the vicinity of the Planck scale. Based on the Hamilton–Jacobi (HJ) equation, by using the corrected classical action, the quantum tunneling method is applied to derive the corrected Hawking temperature within the framework of the generalized uncertainty principle. In the sequel, we check the validity of the first law of thermodynamics and test the thermodynamic instability for the global and local monopole black holes.