  We study the energy-momentum pseudo-tensor of gravitational wave, and examine
the one introduced by Landau-Lifshitz for a general gravitational field and the
effective one recently used in literature. In short wavelength limit after
Brill-Hartle average, both lead to the same gauge invariant stress tensor of
gravitational wave. For relic gravitational waves in the expanding universe, we
examine two forms of pressure, $p_{gw}$ and $\mathcal{P}_{gw}$, and trace the
origin of their difference to a coupling between gravitational waves and the
background matter. The difference is shown to be negligibly small for most of
cosmic expansion stages starting from inflation. We demonstrate that the wave
equation is equivalent to the energy conservation equation using the pressure
$\mathcal{P}_{gw}$ that includes the mentioned coupling.
