  We present a general and unified formulation which can handle the classical
evolution and quantum generation processes of the cosmological gravitational
wave in a broad class of generalized gravity theories. Applications are made in
several inflation models based on the scalar-tensor theory, the induced
gravity, and the low energy effective action of string theory. The
gravitational wave power spectrums based on the vacuum expectation value of the
quantized fluctuating metric during the pole-like inflation stages are derived
in analytic forms. Assuming that the gravity theory transits to Einstein one
while the relevant scales remain in the superhorizon scale, we derive the
consequent power spectrums and the directional fluctuations of the relic
radiation produced by the gravitational wave. The spectrums seeded by the
vacuum fluctuations in the pole-like inflation models based on the generalized
gravity show a distinguished common feature which differs from the scale
invariant spectrum generated in an exponential inflation in Einstein gravity
which is supported by observations.
