Black hole thermodynamics with generalized uncertainty principle

We apply the generalized uncertainty principle to the thermodynamics of a small black hole. Here we have a black hole system with the UV cutoff. It is shown that the minimal length induced by the GUP interrupts the Gross–Perry–Yaffe phase transition for a small black hole. In order to see whether the black hole remnant takes place a transition to a large black hole, we introduce a black hole in a cavity (IR system). However, we fail to show the phase transition of the remnant to the large black hole.