We consider a scalar field which couples to gravity via an arbitrary function of the metric determinant, so that the matter action is no longer invariant under diffeomorphisms (Diff) but only under transverse diffeomorphisms (TDiff). We perform an effective description of the scalar field by treating it as a perfect fluid, consider some limiting cases, and study its energy conditions. Whereas Diff invariance guarantees the conservation of the energy-momentum tensor in General Relativity, for TDiff invariant matter the conservation is satisfied only on solutions to the Einstein equations, and imposes one additional constraint on the metric tensor. Finally, we also consider some particular cases.