In a single composite ferromagnetic/antiferromagnetic (AFM) nanoparticle with an unconventional core/shell morphology, the effects of temperature and interfacial coupling on the exchange bias (EB) are studied by performing a modified Monte Carlo method. With the decrease of temperature, three distinct behaviors of the EB are obtained as a result of the state transitions of the AFM spins from completely free via partially free and partially frozen, and, finally, to completely frozen. The behaviors of the coercivity are nonmonotonic due to the energy competitions and the thermal fluctuations. Moreover, the cooling field affects strongly the low-temperature behaviors of the EB. The EB field obtained after field cooling is negative and almost constant, independent of the interfacial coupling. However, after zero-field cooling, the EB field may exhibit a sign-reversal behavior with increasing interfacial coupling at low temperature. On the basis of the theory of the surplus magnetization, it is revealed that the drastic temperature dependence and the energy competition mechanism at the interface in the cooling process are responsible for this unique property in such novel nanostructures.