  The doubly special relativity (DSR) theories are investigated in order to
take into account an observer-independent length scale in special relativity
framework. It is widely believed that any quantum theory of gravity would
reduce to a DSR model at the flat limit when purely gravitational and quantum
mechanical effects are negligible. Gravity's rainbow is a simple generalization
of DSR theories to incorporate gravity. In this paper, we show that the
effective Friedmann equations that are suggested by loop quantum cosmology
(LQC) can be exactly reobtained in rainbow cosmology setup. The deformed
geometry of LQC then completely fixes the modified dispersion relation and
results in unique DSR model. In comparison with standard LQC scenario where
only the geometry is modified, both of the geometry and matter parts get
modifications in our setup. In this respect, we find that the total number of
microstates for the universe is finite which suggests the statistical origin
for the energy and entropy density bounds. These results explicitly show that
the DSR theories are appropriate candidates for the flat limit of loop quantum
gravity.
