We study the cosmological implications of gravity models which break diffeomorphisms (Diff) invariance down to transverse diffeomorphisms (TDiff). We consider the most general gravitational action involving up to second derivatives of the metric tensor and identify TDiff models as the only stable theories consistent with local gravity tests. The model propagates an additional scalar graviton and although it is indistinguishable from GR in local tests, its cosmological dynamics exhibits a rich phenomenology. Thus we show that the model contains standard LCDM as a solution when the extra scalar mode is not excited, but different cosmological evolutions driven by the new term are possible which we will present in this talk.