Recent observational evidence have established that about 96% of the energy content of the Universe is invisible and made of puzzling ingredients Dark Matter (DM) and Dark Energy (DE). One of the most interesting ways to elaborate a coherent physical description of the whole dark sector is by introducing new direct interactions between DE and DM. However, such interactions make the particle masses of the last varying and therefore constitutes a violation of the weak equivalence principle (WEP), which has consequently to be restricted to ordinary visible matter. Doing so, the validity of the strong equivalence principle (SEP), that extends the WEP to gravitational binding energies, must then also be questionned. The Abnormally Weighting Energy (AWE) hypothesis suggests such an interpretation of DE in terms of a relaxation of WEP on cosmological scales: the laws of gravitation rule both the expansion of space-time and the variation of the gravitational strengths that are experienced differently by the visible and invisible matter sectors. This model mimics the action of an additional fluid with negative pressures although the matter content contains exclusively pressureless matter (ordinary matter and DM). Cosmic acceleration is then reduced to a large-scale manifestation of SEP violation that follows from the repulsive action of the mass-varying DM, while general relativity is still satisfied at low scales, as SEP violation depends on the relative concentration of DM and ordinary matter. We will show how this new interpretation of DE as an anomalous gravity of DM allows to constraint, from supernovae alone, the amount of baryons in the Universe, how it links together the running coupling of gravitation and the varying DM mass, and how it leads to several other remarkable cosmological predictions.