  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.
