  A theory of the quasidilaton is an extension of massive gravity by a scalar
field, nonlinearly realizing a certain new global symmetry of the Lagrangian.
It has been shown that unlike pure massive gravity, this theory does admit
homogeneous and isotropic spatially flat solutions. Among the latter,
selfaccelerated solutions attract a special attention. Previous studies of
perturbations, performed in the decoupling limit, revealed one healthy scalar
mode, while the second relevant scalar was not captured in that limit. Here we
study full cosmological perturbations above the simplest selfaccelerated
background. We show that the fluctuations of a mixed state of the quasidilaton
and the helicity-0 graviton necessarily have a negative kinetic term at short
distances, making this background unphysical. In addition, these cosmologies
exhibit an order one sensitivity to higher dimensional terms suppressed by an
energy scale that is parametrically higher than the strong coupling scale of
the quasidilaton effective theory: such terms include Galileons, Goldstone-like
selfinteractions and derivatives of the quasidilaton coupled to curvature, none
of which introduce extra Ostrogradsky states. As one consequence, cosmology at
the Hubble distances for this particular class of solutions depends on an
unknown extension of the quasidilaton below its strong coupling distance scale.
We note that non-FRW solutions that are similar to those of pure massive
gravity should not necessarily suffer from these pathologies.
