Stability of cosmological solutions in extended quasidilaton massive gravity

We consider the stability of self-accelerating solutions to extended quasidilaton massive gravity in the presence of matter. By making a second metric dynamical in this model, matter can cause it to evolve from a Lorentzian to Euclidean signature, triggering a ghost instability. We study this possibility with scalar field matter as it can model a wide range of cosmological expansion histories. For the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ expansion history, stability considerations substantially limit the available parameter space while for a kinetic-energy-dominated expansion, no choice of quasidilaton parameters is stable. More generally these results show that there is no mechanism intrinsic to the theory to forbid such pathologies from developing from stable initial conditions and that stability can only be guaranteed for particular choices for the matter configuration.