  In this paper we present a construction of effective cosmological models
which describe the propagation of a massive quantum scalar field on a quantum
anisotropic cosmological spacetime. Each obtained effective model is
represented by a rainbow metric in which particles of distinct momenta
propagate on different classical geometries. Our analysis shows that upon
certain assumptions and conditions on the parameters determining such
anisotropic models, we surprisingly obtain a unique deformation parameter
$\beta$ in the modified dispersion relation of the modes. Hence inducing an
isotropic deformation despite the general starting considerations. We then
ensure the recovery of the dispersion relation realized in the isotropic case,
studied in [arXiv:1412.6000], when some proper symmetry constraints are
imposed, and we estimate the value of the deformation parameter for this case
in loop quantum cosmology context.
