  There is a no-go theorem forbidding flat and closed FLRW solutions in massive
gravity on a flat reference metric, while open solutions are unstable. Recently
it was shown that this no-go theorem can be overcome if at least some matter
couples to a hybrid metric composed of both the dynamical and the fixed
reference metric. We show that this is not compatible with the standard
description of cosmological sources in terms of effective perfect fluids, and
the predictions of the theory become sensitive either to the detailed
field-theoretical modelling of the matter content or to the presence of
additional dark degrees of freedom. This is a serious practical complication.
Furthermore, we demonstrate that viable cosmological background evolution with
a perfect fluid appears to require the presence of fields with highly contrived
properties. This could be improved if the equivalence principle is broken by
coupling only some of the fields to the composite metric, but viable
self-accelerating solutions due only to the massive graviton are difficult to
obtain.
