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In this work, we investigate the phase transition of the Schwarzschild blackhole (SBH) inside an isothermal spherical cavity in the context of thenon-commutative (NC) gauge theory of gravity, by using the Seiberg-Witten (SW)map and the star product. Firstly, we compute the NC correction to the Hawkingtemperature and derive the logarithmic correction to the entropy, then wederive the local temperature and local energy of NC SBH in isothermal cavity.Our results show that the non-commutativity removes the commutative divergencebehavior of temperature, and prevents the SBH from the complete evaporation,which leads to a remnant black hole, and this geometry has predicted a minimallength in the order of Planck scale $\Theta\sim l_{planck}$. Therefore, thethermodynamic stability and phase transition is studied by analyzing thebehavior of the local heat capacity and the Helmholtz free energy in the NCspacetime, where the results show that, the NC SBH has a two second-order phasetransition and one first-order phase transition, with two Hawking-Page phasetransition in the NC gauge theory.

Schwarzschild black hole surrounded by a cavity and phase transition in the non-commutative gauge theory of gravity