We study the possibility to obtain cosmological late-time acceleration from a geometry changing with the scale, in particular, in the so-called multifractional theories with $q$-derivatives and with weighted derivatives. In the theory with $q$-derivatives, the luminosity distance is the same as in general relativity and, therefore, geometry cannot act as dark energy. In the theory with weighted derivatives, geometry alone is able to sustain a late-time acceleration phase without fine tuning, while being compatible with structure-formation and big-bang nucleosynthesis bounds. This suggests to extend the theory, in a natural way, from just small-scale to also large-scale modifications of gravity. Surprisingly, the Hausdorff dimension of spacetime is constrained to be close to the topological dimension 4. After arguing that this finding might not be a numerical coincidence, we conclude that present-day acceleration could be regarded as the effect of a new restoration law for spacetime geometry.

17 pages, 7 figures, 2 tables. v2: discussion improved, results unchanged, typos corrected