  In this paper we construct higher-dimensional minimal theory of mass-varying
massive gravity (MTMVMG) where the masslike scalar potential is coupled to a
vielbein potential, unlike in the previous literature where it is coupled to
metric, such that the number of graviton degrees of freedom in the theory is
the same as in general relativity. We then study the cosmological aspects of
this theory and show that it has eight critical points: five in the massless
sector and three in the massive sector. In contrast to the standard theory of
mass-varying massive gravity where the graviton mass asymptotically approaches
zero at late times, hence making the contribution of massive gravity to the
late-time cosmic expansion minimum, the MTMVMG can provide good descriptions
both in the massless and massive sectors. Especially, there are at least two
interesting possible scenarios for the late-time cosmology in the theory: the
dark energy is either due to the constant graviton mass which comes from the
scalar field that becomes frozen after the reheating era, or due to the
quintessence paradigm where the scalar field is dynamic. Therefore, if the
accelerating expansion of the universe in the massless sector can be explained
by standard quintessence paradigm, in the massive sector it has to be explained
by the nontrivial interplay between quintessence and massive gravity.
