The observational evidence for the presence of dark matter is now generally accepted, with no lack of possible candidates (see e.g. Dekel, Einasto, and Rees 1986). The proposed candidates are devided into two groups, baryonic and non-baryonic. The latter is further devided to hot and cold dark matter. For the cold dark matter, among the first to be proposed is the axion. In this paper we shall not dwell on numerous cold dark matter candidates offered by particle physicists, for there are review articles on the subject (see e.g. Turner 1986, Primack 1986). The main purpose of the present report is to suggest that neutron star cooling theory and future space satellite programs (e.g. AXAF, XAO, LXAO) have a potential for offering the best astrophysical constraint on the axion mass and hence, giving valuable insight to some cosmological problems.