The abnormally weighting energy hypothesis consists of assuming that the dark sector of cosmology violates the weak equivalence principle (WEP) on cosmological scales, which implies a violation of the strong equivalence principle for ordinary matter. In this paper, dark energy is shown to result from the violation of WEP by pressureless (dark) matter. This allows us to build a new cosmological framework in which general relativity is satisfied at low scales, as WEP violation depends on the ratio of the ordinary matter over dark matter densities, but at large scales, we obtain a general relativity-like theory with a different value of the gravitational coupling. This explanation is formulated in terms of a tensor-scalar theory of gravitation without WEP for which there exists a revisited convergence mechanism toward general relativity. The consequent dark energy mechanism build upon the anomalous gravity of dark matter (i) does not require any violation of the strong energy condition p