  We introduce the Abnormally Weighting Energy (AWE) hypothesis in which dark
energy (DE) is presented as a consequence of the violation of the weak
equivalence principle (WEP) at cosmological scales by some dark sector. Indeed,
this implies a violation of the strong equivalence principle (SEP) for ordinary
matter and consequent cosmic acceleration in the observable frame as well as
variation of the gravitational constant. The consequent DE mechanism build upon
the AWE hypothesis (i) does not require a violation of the strong energy
condition $p<-\rho c^2/3$, (ii) assumes rather non-negligible direct couplings
to the gravitational scalar field (iii) offers a natural convergence mechanism
toward general relativity (iv) accounts fairly for supernovae data from various
couplings and equations of state of the dark sector as well as density
parameters very close to the ones of the concordance model $\Lambda CDM$.
Finally (v), this AWE mechanism typically ends up with an Einstein-de Sitter
expansion regime once the attractor is reached.
