  We investigate the propagation of the gravitational waves in a cosmological
background. Based on the framework of spatially covariant gravity, we derive
the general quadratic action for the gravitational waves. The spatial
derivatives of the extrinsic curvature and the parity-violating terms are
systematically introduced. Special attention is paid to the propagation speed
of the gravitational waves. We find that it is possible to make the two
polarization modes propagate in the same speed, which may differ from that of
the light, in the presence of parity-violating terms in the action. In
particular, we identify a large class of spatially covariant gravity theories
with parity violation, in which both the polarization modes propagate in the
speed of light. Our results imply that there are more possibilities in the
framework of spatially covariant gravity in light of the propagation speed of
the gravitational waves.
