  After neutrinos decouple from the photon bath, they can populate a thermal
dark sector. If this occurs at a temperature above ~100 keV, this can have
measurable impacts on light element abundances. We calculate light element
abundances in this scenario, studying the impact from rapid cooling of the
Standard Model neutrinos, and from an increase in the number of relativistic
degrees of freedom $N_{\rm{eff}}$, which can occur in the presence of a mass
threshold. We incorporate these changes in the publicly available BBN code
PRIMAT, using the reaction networks from PRIMAT and from the BBN code
PArthENoPE, to calculate Y$_{\rm{P}}$ and D/H. We provide limits from the two
different reaction networks as well as with expanded errors to include both
results. If electron neutrinos significantly participate in the cooling, we
find limits down to temperatures as low as 100 keV. If electron neutrinos are
weakly participating (for instance if only the mass eigenstate $\nu_3$
equilibrates), cooling places no limits. However, if the dark sector undergoes
a "step" in $N_{\rm{eff}}$, there can be additional, $\omega_b$-dependent
constraints. These limits can vary from strong (for low values of $\omega_b$)
to a mild preference for new physics (for high values of $\omega_b$). Future
analyses including upcoming CMB data should improve these limits.
