Rainbow black hole thermodynamics and the generalized uncertainty principle

We study the phase transition of rainbow inspired higher dimensional Schwarzschild black hole incorporating the effects of the generalized uncertainty principle. First, we obtain the relation between the mass and Hawking temperature of the rainbow inspired black hole taking into account the effects of the modified dispersion relation and the generalized uncertainty principle. The heat capacity is then computed from this relation which reveals that there are remnants. The entropy of the black hole is next obtained in $3+1$ and $4+1$-dimensions and is found to have logarithmic corrections only in $3+1$-dimensions. We further investigate the local temperature, free energy and stability of the black hole in an isothermal cavity. From the analysis of the free energy, we find that there are two Hawking-Page type phase transitions in $3+1$ and $4+1$-dimensions if we take into account the generalized uncertainty principle. However, in the absence of the generalized uncertainty principle, only one Hawking-Page type phase transition exists in spacetime dimensions greater than four.