The problem of cosmological production of gravitational waves is discussed in the framework of an expanding, spatially homogeneous and isotropic FRW type Universe with time-evolving vacuum energy density. The gravitational wave equation is established and its modified time-dependent part is analytically resolved for different epochs in the case of a flat geometry. Unlike the standard $Lambda$CDM cosmology (no interacting vacuum), we show that gravitational waves are produced in the radiation era even in the context of general relativity. We also show that for all values of the free parameter, the high frequency modes are damped out even faster than in the standard cosmology both in the radiation and matter-vacuum dominated epoch. The formation of the stochastic background of gravitons and the remnant power spectrum generated at different cosmological eras are also explicitly evaluated. It is argued that measurements of the CMB polarization (B-modes) and its comparison with the rigid $Lambda$CDM model plus the inflationary paradigm may become a crucial test for dynamical dark energy models in the near future.
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