Is Dark Energy Abnormally Weighting?

We present a new interpretation of dark energy in terms of an \textit{Abnormally Weighting Energy} (AWE). This means that dark energy does not couple to gravitation in the same way as ordinary matter, yielding a violation of the weak and strong equivalence principles on cosmological scales. The resulting cosmological mechanism accounts for the Hubble diagram of type Ia supernovae in terms of both cosmic acceleration and variation of the gravitational constant while still accounting for the present tests of general relativity. This explanation allows to build dark energy models (i) without violation of the strong energy condition $p<-\rho c^2/3$ (ii) with non-negligible direct couplings to gravitation and (iii) natural convergence mechanism toward general relativity.