  We compute the compressibility of the conduction electrons in both bulk
orthorhombic FeSe and monolayer FeSe on SrTiO$_3$ substrate, including
dynamical electronic correlations within slave-spin mean-field +
density-functional theory. Results show a zone of enhancement of the electronic
compressibility crossing the interaction-doping phase diagram of these
compounds in accord with previous simulations on iron pnictides and in general
with the phenomenology of Hund's metals. Interestingly at ambient pressure FeSe
is found slightly away from the zone with enhanced compressibility but moved
right into it with hydrostatic pressure, while in monolayer FeSe the stronger
enhancement region is realized on the electron-doped side. These findings
correlate positively with the enhancement of superconductivity seen in
experiments, and support the possibility that Hund's induced many-body
correlations boost superconductive pairing when the system is at the frontier
of the normal- to Hund's-metal crossover.
