  We re-examine a recently proposed scenario where the deformed dispersion
relations associated with a flow of the spectral dimension to a UV value of 2
leads to a scale-invariant spectrum of cosmological fluctuations, without the
need for inflation. In that scenario Einstein gravity was assumed. The theory
displays a wavelength-dependent speed of light but by transforming to a
suitable "rainbow frame" this feature can be removed, at the expense of
modifying gravity. We find that the ensuing rainbow gravity theory is such that
gravity switches off at high energy (or at least leads to a universal conformal
coupling). This explains why the fluctuations are scale-invariant on all
scales: there is no horizon scale as such. For dispersion relations that do not
lead to exact scale invariance we find instead esoteric inflation in the
rainbow frame. We argue that these results shed light on the behaviour of
gravity under the phenomenon of dimensional reduction.
