  Light dark matter is a compelling experimental target in light of stringent
constraints on heavier WIMPs. However, for a sub-MeV WIMP, the universe is
sufficiently well understood at temperatures below 10 MeV that there is no room
for it to be a thermal relic. Avoiding thermalization is itself a strong
constraint with significant implications for direct detection. In this paper,
we explore the space of models of sub-MeV dark matter with viable cosmologies.
The parameter space of these models that is also consistent with astrophysical
and lab-based limits is highly restricted for couplings to electrons but
somewhat less constrained for nuclei. We find that achieving nuclear
cross-sections well-above the neutrino floor necessarily predicts a new
contribution to the effective number of neutrino species, $\Delta N_{\rm eff} =
0.09$ that will be tested by the next generation of CMB observations. On the
other hand, models with absorption signatures of dark matter are less
restricted by cosmology even with future observations.
