  The properties of gravitational-wave (GW) propagation are modified in
alternative theories of gravity and are crucial observables to test gravity at
cosmological distance. The propagation speed has already been measured from
GW170817 so precisely and pinned down to the speed of light, while other
properties of GW propagation have not constrained tightly yet. In this paper,
we investigate the measurement precisions of the amplitude damping rate
(equivalently, the time variation of the gravitational coupling for GWs) and
graviton mass in the generalized framework of GW propagation with the future
detectors such as Voyager, Cosmic Explorer, and Einstein Telescope. As a
result, we show that the future GW observation can reach 1% error for the
amplitude damping. We also study the time variation of the gravitational
couplings in Horndeski theory by performing Monte Carlo-based numerical
simulations. From the simulation results, we find that the current accelerating
Universe prefers the models with less damping of GWs and that the equivalence
principle can be tested at the level of 1% by the future GW observation.
