  We consider the possibility that the cosmic neutrino background might have a
nonthermal spectrum, and investigate its effect on cosmological parameters
relative to standard $\Lambda$-Cold Dark Matter ($\Lambda$CDM) cosmology. As a
specific model, we consider a thermal $y$-distortion, which alters the
distribution function of the neutrino background by depleting the population of
low-energy neutrinos and enhancing the high-energy tail. We constrain the
thermal $y$-parameter of the cosmic neutrino background using Cosmic Microwave
Background (CMB) and Baryon Acoustic Oscillation (BAO) measurements, and place
a $95\%$-confidence upper bound of $y \leq 0.043$. The $y$-parameter increases
the number of effective relativistic degrees of freedom, reducing the sound
horizon radius and increasing the best-fit value for the Hubble constant $H_0$.
We obtain an upper bound on the Hubble constant of $H_0 = 71.12\
\mathrm{km/s/Mpc}$ at $95\%$ confidence, substantially reducing the tension
between CMB/BAO constraints and direct measurement of the expansion rate from
Type-Ia supernovae. Including a spectral distortion also allows for a higher
value of the spectral index of scalar fluctuations, with a best-fit of
$n_{\mathrm{S}} = 0.9720 \pm 0.0063$, and a $95\%$-confidence upper bound of
$n_{\mathrm{S}} \leq 0.9842$.
