  We compute the gravitational leptogenesis generated from the parity-violating
gravitational waves sourced by an abelian gauge field coupled to a
pseudo-scalar inflation. We show that, once the CMB bound on the
tensor-to-scalar ratio is enforced, the lepton asymmetry produced by this
mechanism during inflation is too small to account for the observed baryon
asymmetry of the universe, irrespectively of the inflaton potential, the
strength of its coupling to the gauge field, and the details of reheating.
