We study the enhanced spin relaxation time of Au nanoparticles in nanopillar-shaped double-barrier junction devices with a stacked Fe/MgO/Au-nanoparticle/MgO/Fe structure. The size of Au nanoparticles located in a current path is deduced from a transmission electron micrograph and the Coulomb blockade behavior in the current-voltage characteristics of the devices. A finite tunnel magnetoresistance (TMR) is observed above a critical current and is attributable to spin accumulation in Au nanoparticles. Based on a simple model of TMR due to spin accumulation in a nanoparticle, the spin relaxation time tau sub()sis estimated from the magnitude of the critical current. The temperature and bias-voltage region where TMR appears are determined from systematic observations, showing that the appearance of TMR is not associated with the Coulomb blockade but with spin accumulation. We find that the obtained tau sub()sis anomalously extended (~800 ns) at low temperatures and abruptly decreases above a critical temperature. Interestingly, the critical temperature strongly depends on the size of the Au nanoparticles and is much lower than the effective temperature corresponding to the discrete energy spacing. A theoretical analysis for the spin relaxation of electrons with discrete energy levels shows that not only the anomalously extended spin relaxation time, but also the strong temperature dependence of tau sub()sarise from the broadening of discrete energy levels due to coupling with phonons in the surrounding matrix. Numerical calculations using reasonable parameter values well reproduce the observed temperature and size dependence of the spin relaxation time in Au nanoparticles.