  We review a cosmological model where the metric determinant plays a dynamical
role and present new numerical results on the cancellation of the vacuum energy
density including the contribution of a cosmological constant. The action of
this model is only invariant under restricted coordinate transformations with
unit Jacobian (the same restriction appears in the well-known
unimodular-gravity approach to the cosmological constant problem). As to the
possible origin of the nonstandard terms in the matter action of the model, we
show that these terms can, in principle, arise from the emergent gravity in the
IIB matrix model, a nonperturbative formulation of superstring theory.
