  In this paper we shall investigate the propagation of gravitational waves in
a flat Friedman-Robertson-Walker background, in the context of a string
motivated corrected Einstein gravity. Particularly, we shall consider a
misalignment axion Einstein gravity in the presence of a string originating
Chern-Simons coupling of the axion field to the Chern-Pontryagin density in
four dimensions. We shall focus our study on the propagation of the
gravitational waves, and we shall investigate whether there exists any
difference in the propagation of the polarization states of the gravitational
waves. As we demonstrate, the dispersion relations are different in the
Right-handed mode and the Left-handed mode. Finally, we compare the propagation
of the axion Chern-Simons Einstein theory with that of standard $F(R)$ gravity.
