Cosmology in Weyl Transverse Gravity

We study the Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmology in the Weyl-transverse (WTDiff) gravity in a general space-time dimension. The WTDiff gravity is invariant under both the local Weyl (conformal) transformation and the volume preserving diffeormorphisms (transverse diffeomorphisms) and is believed to be equivalent to general relativity at least at the classical level (perhaps, even in the quantum regime). It is explicitly shown by solving the equations of motion that the FLRW metric is a classical solution in the WTDiff gravity only when the spatial metric is flat, that is, the Euclidean space, and the lapse function is a nontrivial function of the scale factor.