  The linear relation between the entropy and area of a black hole can be
derived from the Heisenberg principle, the energy-momentum dispersion relation
of special relativity, and general considerations about black holes. There
exist results in quantum gravity and related contexts suggesting the
modification of the usual dispersion relation and uncertainty principle. One of
these contexts is the gravity's rainbow formalism. We analyze the consequences
of such a modification for black hole thermodynamics from the perspective of
two distinct rainbow realizations built from doubly special relativity. One is
the proposal of Magueijo and Smolin and the other is based on a canonical
implementation of doubly special relativity put forward recently by the
authors. In these scenarios, we obtain modified expressions for the entropy and
temperature of black holes. We show that, for a family of doubly special
relativity theories satisfying certain properties, the temperature can vanish
in the limit of zero black hole mass. For the Magueijo and Smolin proposal,
this is only possible for some restricted class of models with bounded energy
and unbounded momentum. With the proposal of a canonical implementation, on the
other hand, the temperature may vanish for more general theories; in
particular, the momentum may also be bounded, with bounded or unbounded energy.
This opens new possibilities for the outcome of black hole evaporation in the
framework of a gravity's rainbow.
