  We present a set of equations describing the cosmological gravitational wave
in a gravity theory with quadratic order gravitational coupling terms which
naturally arise in quantum correction procedures. It is known that the
gravitational wave equation in the gravity theories with a general $f(R)$ term
in the action leads to a second order differential equation with the only
correction factor appearing in the damping term. The case for a $R^{ab} R_{ab}$
term is completely different. The gravitational wave is described by a fourth
order differential equation both in time and space. However, curiously, we find
that the contributions to the background evolution are qualitatively the same
for both terms.
