  By presenting a relation between average energy of the ensemble of probe
photons and energy density of the Universe, in the context of {\it gravity's
rainbow} or {\it doubly general relativity} scenario, we introduce a rainbow
FRW Universe model. By analyzing the fixed points in flat FRW model modified by
two well known rainbow functions, we find that the finite time singularity
avoidance (i.e. Big-Bang) may still remain as a problem. Then, we follow the
"Emergent Universe" scenario in which there is no beginning of time and
consequently there is no Big-Bang singularity. Moreover, we study the impact of
a high energy quantum gravity modifications related to the gravity's rainbow on
the stability conditions of an "Einstein static Universe" (ESU). We find that
independent of a particular rainbow function, the positive energy condition
dictates a positive spatial curvature for the Universe. In fact, without
raising a nonphysical energy condition in the quantum gravity regimes, we can
address an agreement between gravity's rainbow scenario and basic assumption of
modern version of "Emergent Universe". We show that in the absence and presence
of an energy-dependent cosmological constant $\Lambda(\epsilon)$, a stable
Einstein static solution is available versus the homogeneous and linear scalar
perturbations under the variety of obtained conditions. Also, we explore the
stability of ESU against the vector and tensor perturbations.
