  Axion dark matter is interesting as it allows a natural coupling to the
gravitational Chern-Simons term. In the presence of an axion background, the
gravitational Chern-Simons term produces parity violating effects in the
gravitational sector, in particular on the propagation of gravitational waves.
Previously, it has been shown that the coherent oscillation of the axion field
leads to a parametric amplification of gravitational waves with a specific
frequency. In this paper we focus on the parity violating effects of the
Chern-Simon coupling and show the occurrence of gravitational birefringence. We
also find deviation from the speed of light of the velocity of the
gravitational waves. We give constraints on the axion-Chern-Simons coupling
constant and the abundance of the axion dark matter from the observation of
GW170817 and GRB170817A.
