  The Cosmic Neutrino Background (C$\nu$B) constitutes the last observable
prediction of the standard cosmological model, which has yet to be detected
directly. In this work, we show how the coherent scattering of neutrinos off
dense neutron matter can lead to an additional cooling channel in neutron stars
(NSs). We also include the effects of gravitational capture and boosting, but
find that the cooling is efficient only in the presence of large overdensities.
We further discuss the impact of boosted C$\nu$B flux on Earth from nearby NSs
and the potential detection prospects in the case of a future nearby galactic
supernova. Although currently these ideas do not offer any detection prospects,
they can be used to constrain overdensities on short length scales
$\mathcal{O}(10\text{ km})$. We also discuss the impact of new physics
scenarios, such as long-range forces, on neutron star cooling through the
C$\nu$B.
