  We study the cosmological implications of gravity models which break
diffeomorphisms (Diff) invariance down to transverse diffeomorphisms (TDiff).
We start from the most general gravitational action involving up to quadratic
terms in derivatives of the metric tensor and identify TDiff models as the only
stable theories consistent with local gravity tests. These models propagate an
additional scalar graviton and although they are indistinguishable from GR at
the post-Newtonian level, their cosmological dynamics exhibits a rich
phenomenology. Thus we show that the model includes standard $\Lambda$CDM as a
solution when the extra scalar mode is not excited, but different cosmological
evolutions driven by the new term are possible. In particular, we show that for
a soft Diff breaking, the new contribution always behaves as a cosmological
constant at late times. When the extra contribution is not negligible,
generically its evolution either behaves as dark energy or tracks the dominant
background component. Depending on the initial conditions, solutions in which
the universe evolves from an expanding to a contracting phase, eventually
recollapsing are also possible.
