  A new class of modified theory of gravity is introduced where the volume form
becomes dynamical. This approach is motivated by unimodular gravity and can
also be related to Brans-Dicke theory. On the level of the action, the only
change made will be through the volume element which is used in the
integration. This is achieved by the introduction of a fourth order tensor
which connects the spacetime metric to the new volume form. Using dynamical
systems techniques, this model is studied in the context of cosmology. The most
interesting result is that there exist parameter ranges where this model starts
undergoing an epoch of accelerated expansion, followed by a decelerating
expansion which evolves to a final epoch of accelerated expansion.
