Abstract
In this letter, we consider the possibility of reconciling metric theories of gravitation with violation of the conservation of energy-momentum. Under some circumstances, this can be achieved in the context of unimodular gravity, and it leads to the emergence of an effective cosmological constant in Einstein's equation. We specifically investigate two potential sources of energy non-conservation ---non-unitary modifications of quantum mechanics, and phenomenological models motivated by quantum gravity theories with spacetime discreteness at the Planck scale--- and show that such locally negligible phenomena can nevertheless become relevant at the cosmological scale.