  Using a one-dimensional minisuperspace model with a dimensionless ratio
$\frac{E}{E_{Pl}}$, we study the initial singularity problem at the quantum
level for the closed rainbow cosmology with a homogeneous, isotropic classical
space-time background. We derive the classical Hamiltonian within the framework
of Schutz's formalism for an ideal fluid with a cosmological constant. We
characterize the behavior of the system at the early stages of the universe
evolution through analyzing the relevant shapes for the potential sector of the
classical Hamiltonian for various matter sources, each separately modified by
two rainbow functions. We show that for both rainbow universe models presented
here, there is the possibility of eliminating the initial singularity by
forming a potential barrier and static universe for a non-zero value of the
scale factor. We investigate their quantum stability and show that for an
energy-dependent space-time geometry with energies comparable with the Planck
energy, the non-zero value of the scale factor may be stable. It is shown that
under certain constraints the rainbow universe model filled with an exotic
matter as a domain wall fluid plus a cosmological constant can result in a
non-singular harmonic universe. In addition, we demonstrate that the
harmonically oscillating universe with respect to the scale factor is sensitive
to $\frac{E}{E_{Pl}}$ and that at high energies it may become stable quantum
mechanically. Through a Schr\"{o}dinger-Wheeler-De Witt (SWD) equation obtained
from the quantization of the classical Hamiltonian, we also extract the wave
packet of the universe with a focus on the early stages of the evolution.
