  Gravitons produced from quantum vacuum fluctuations during an inflationary
stage in the early Universe have zero entropy as far as they reflect the time
evolution (squeezing) of a pure state, their large occupation number
notwithstanding. A non-zero entropy of the gravitons (classical gravitational
waves (GW) after decoherence) can be obtained through coarse graining. The
latter has to be physically justified {\it and} should not contradict
observational constraints. We propose two ways of coarse graining for which the
fixed temporal phase of each Fourier mode of the GW background still remains
observable: one based on quantum entanglement, and another one following from
the presence of a secondary GW background. The proposals are shown to be
mutually consistent. They lead to the result that the entropy of the primordial
GW background is significantly smaller than it was thought earlier. The
difference can be ascribed to the information about the regular (inflationary)
initial state of the Universe which is stored in this background and which
reveals itself, in particular, in the appearance of primordial peaks (acoustic
peaks in the case of scalar perturbations) in the multipole spectra of the CMB
temperature anisotropy and polarization.
