  We define a geometric quantity for asymptotically hyperbolic manifolds, which
we call the volume-renormalized mass. It is essentially a linear combination of
the ADM mass surface integral and a renormalization of the volume.
  We show that the volume-renormalized mass is well-defined and diffeomorphism
invariant under weaker fall-off conditions than required to ensure that the
renormalized volume and the ADM mass surface integral are well-defined
separately. We prove several positivity results for the volume-renormalized
mass. We also use it to define a renormalized Einstein--Hilbert action and a
renormalized expander entropy which is nondecreasing under the Ricci flow.
Further, we show that local maximizers of the entropy are local minimizers of
the volume-renormalized mass.
