  The thermal evolution of a neutron star is studied by including the energy
loss due to axion emission. Two axion models and three types of neutron-star
matter equation of state are used with the effects of nucleon superfluidity
properly taken into account. In comparison with the observational data of
PSR0656+14 from ROSAT, the upper limits on the axion mass are found to be $m_a
< 0.06 - 0.3$ eV and 0.08$ -$ 0.8 eV for the KSVZ and DFSZ axion models,
respectively, with the soft equation of state giving the most stringent limits.
