We argue that a cosmic neutrino background that carries non-zero lepton charge develops gravitational instabilities. Fundamentally, these instabilities are related to the mixed gravity-lepton number anomaly. We have explicitly computed the gravitational Chern-Simons term which is generated quantum-mechanically in the effective action in the presence of a lepton number asymmetric neutrino background. The induced Chern-Simons term has a twofold effect: (i) gravitational waves propagating in such a neutrino background exhibit birefringent behaviour leading to an enhancement/suppression of the gravitational wave amplitudes depending on the polarisation, where the magnitude of this effect is related to the size of the lepton asymmetry; (ii) Negative energy graviton modes are induced in the high frequency regime, which leads to very fast vacuum decay of a vacuum state into, e.g., positive energy photons and negative energy gravitons. From the constraint on the present radiation energy density, we obtain an interesting bound on the lepton asymmetry of the universe.