  In this paper, we investigate the thermodynamic properties of black holes in
the framework of rainbow gravity. By considering rainbow functions in the
metric of Schwarzschild and Reissner-Nordstr\"{o}m black holes, remnant and
critical masses are found to exist. Demanding the universality of logarithmic
corrections to the semi-classical area law for the entropy leads to
constraining the form of the rainbow functions. The mass output and the
radiation rate for these constrained form of rainbow functions have been
computed for different values of the rainbow parameter $\eta$ and have striking
similarity to those derived from the generalized uncertainty principle.
