  Gravitational counterpart of the chiral magnetic effect, which is referred as
the chiral gravitational effect, can also be of interest in a cosmological
setup. In this study, we investigate this effect in the time-dependent chiral
asymmetric fermion background and in the expanding spacetime by formulating the
effective action of gravitational waves. We also analyze the anomaly equation
to see how the backreaction from gravitational waves to thermal chiral plasma
occurs. We find that the non-trivial time dependence of chiral chemical
potential, which can be induced in some scenarios of baryogenesis, is the key
ingredient of the chiral gravitational effect. It turns out that the "memory"
of the effect is imprinted on the high frequency gravitational waves
propagating in the plasma. Cosmological implications and potential effects on
the gravitational wave observation are briefly discussed.
