  We reanalyze the behavior of Friedmann-Lema\^itre-Robertson-Walker
cosmologies in the recently proposed quasidilaton massive-gravity model, and
discover that the background dynamics present hitherto unreported features that
require unexpected fine-tuning of the additional fundamental parameters of the
theory for an observationally consistent background cosmology. We also identify
new allowed regions in the parameters space and exclude some of the previously
considered ones. The evolution of the mass of gravitational waves reveals
non-trivial behavior, exhibiting a mass squared that may be negative in the
past, and that presently, while positive, is larger than the square of the
Hubble parameter. These properties of the gravity-wave mass have the potential
to lead to observational tests of the theory. While quasidilaton massive
gravity is known to have issues with stability at short distances, the current
analysis is a first step toward the investigation of the more stable extended
quasidilaton massive gravity theory, with some expectation that both the
fine-tuning of parameters and the interesting behavior of the gravity-wave mass
will persist.
