  Thermal QCD equations of state at high baryon density are sensitive to the
phase structure and the resulting excitation modes. The leading contribution at
low temperature can be either ~p_F^2 T^2 (pF: Fermi momentum, T: temperature)
for phases with gapless quarks, or ~T^4 for phases with gapped quarks. In the
latter the thermal pressure is dominated by collective modes. Starting with a
schematic quark model developed for neutron star structure, we estimate the
thermal contributions and zero point energy from the Nambu-Goldstone modes by
building them upon the mean field background for the color-flavor-locked quark
matter. Applying the phase shift representation for thermodynamic potentials,
we include not only the bound state pairs but also resonating pairs. According
to the Levinson's theorem, the high energy contributions tend to cancel the
pole contributions to the thermodynamics, tempering the UV behaviors in the
zero point energy. Our primary target in this talk is the domain with baryon
density nB as large as ~ 5-10n_0 (n_0 = 0.16 fm^{-3}: nuclear saturation
density), and the temperature T of the order ~30-100 MeV. The insights into
this domain may be obtained through the future detection of gravitational waves
from neutron star merging events.
