  Unimodular gravity addresses the old cosmological constant (CC) problem,
explaining why such constant is not at least as large as the largest particle
mass scale, but classically it is indistinguishable from ordinary gravity.
Conversely, quantum physics may give us a way to distinguish the two theories.
Thus, here the unimodular constraint is imposed on a non-perturbative and
background-independent quantum version of quadratic gravity, which was recently
formulated. It is shown that unimodularity does lead to different predictions
for some inflationary quantum observables. Unimodular gravity per se does not
solves the new CC problem (why the CC has the observed value?) even in this
realization. To address this issue a multiverse made by different eras in a
single big bang is considered and the observed scale of dark energy is
explained anthropically.
