  We examine the gravitational wave frequencies from neutron stars during
thermal evolution, adopting the relativistic Cowling approximation. We
particularly focus on the neutron star models, in which the direct Urca (rapid
cooling process) does not work, without the superfluidity and
superconductivity. For such models, the cooling curve hardly depends on the
equation of state (EOS) as well as the mass of neutron star, while we show that
the gravitational wave frequencies strongly depend on the both properties.
Then, we find that the frequencies of the fundamental and the 1st pressure mode
multiplied with the stellar mass are well expressed as a function of the
stellar compactness almost independently of the EOS. We also find that the
frequency of the 1st gravity mode in later phase of the thermal evolution is
strongly correlated with the stellar compactness. In addition, we derive the
empirical formula estimating the threshold mass for the onset of the direct
Urca inside the neutron star as a function of the nuclear saturation parameter.
This formula will give us a constraint on the neutron star properties, if it
would be observationally found that the direct Urca occurs (or does not work)
inside the neutron star.
