  We investigate the cosmological evolution in a universe governed by the
extended, varying-mass, nonlinear massive gravity, in which the graviton mass
is promoted to a scalar-field. We find that the dynamics, both in flat and open
universe, can lead the varying graviton mass to zero at late times, offering a
natural explanation for its hugely-constrained observed value. Despite the
limit of the scenario towards standard quintessence, at early and intermediate
times it gives rise to an effective dark energy sector of a dynamical nature,
which can also lie in the phantom regime, from which it always exits naturally,
escaping a Big-Rip. Interestingly enough, although the motivation of massive
gravity is to obtain an IR modification, its varying-mass extension in
cosmological frameworks leads to early and intermediate times modification
instead.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics