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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 $$n_{\mathrm{S}} \leq 0.9842$$.