Hello again,
It appears that the idea for 'Rainbow Gravity' was first set out in a paper by Magueijo and Smolin called Gravity's Rainbow that appeared in 'Classical and Quantum Gravity' in '04. The most important piece of that work to the current effort and the 'Dimensional Reduction..' paper is found in the following excerpt.
"the geometry of spacetime may depend on the energy of a particle moving in it. Thus, spacetime geometry has an effective description; in the language of the renormalization group, geometry "runs." Hence there is no single spacetime dual to momentum space; the dual to momentum space is the energy dependent family of metrics."
That is; the spectral dimension D runs with the energy E we use, or equivalently with the smallness of scale we probe the dimensionality of space in, making spacetime appear 2-d as we approach a UV cutoff (toward the Planck scale) and 4-d toward the infrared, or as we approach the common level of scale. The 'family of metrics' noted in the quote above constitute the 'rainbow' of Rainbow Gravity.
More later,
Jonathan