Using a one-dimensional minisuperspace model with a dimensionless ratio 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 E/E_{Pl} and that at high energies it may become stable quantum mechanically. Through a Schrödinger-Wheeler-De Witt 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. The resulting wave packet supports the existence of a bouncing non-singular universe within the context of gravity's rainbow proposal.
