  We analyze multi-orbital Hubbard models describing Hund's metals, focusing on
the ubiquitous occurrence of a charge instability, signalled by a
divergent/negative electronic compressibility, in a range of doping from the
half-filled Mott insulator corresponding to the frontier between Hund's and
normal metals. We show that the breaking of rotational invariance favors this
instability: both spin-anisotropy in the interaction and crystal-field
splitting among the orbitals make the instability zone extend to larger
dopings, making it relevant for real materials like iron-based superconductors.
These observations help us build a coherent picture of the occurrence and
extent of this instability. We trace it back to the partial freezing of the
local degrees of freedom in the Hund's metal, which reduces the allowed local
configurations and thus the quasiparticle itinerancy. The abruptness of the
unfreezing happening at the Hund's metal frontier can be directly connected to
a rapid change in the electronic kinetic energy and thus to the enhancement and
divergence of the compressibility.
