The temperature dependence of the topological susceptibility in QCD, chi_t, essentially determines the abundance of the QCD axion in the Universe, and is commonly estimated, based on the instanton picture, to be a certain negative power of temperature. While lattice QCD should be able to check this behavior in principle, the temperature range where lattice QCD works is rather limited in practice, because the topological charge is apt to freezes at high temperatures. In this work, two exploratory studies are presented. In the first part, we try to specify the temperature range in the quenched approximation. Since our purpose here is to estimate the range expected in unquenched QCD through quenched simulations, hybrid Monte Carlo (HMC) algorithm is employed instead of heatbath algorithm. We obtain an indication that unquenched calculations of chi_t encounter the serious problem of autocorrelation already at T~2Tc or even below with the plain HMC. In the second part, we revisit the axion abundance. The absolute value and the temperature dependence of chi_t in real QCD can be significantly different from that in the quenched approximation, and is not well established above the critical temperature. Motivated by this fact and precedent arguments which disagree with the conventional instanton picture, we estimate the axion abundance in an extreme case where chi_t decreases much faster than the conventional power-like behavior. We find a significant enhancement of the axion abundance in such a case.
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