We have studied the temperature dependence of the exchange bias effect in epitaxial Co/CoO bilayer structures with in-plane uniaxial magnetocrystalline anisotropy. We have measured the anisotropic positive exchange bias, which is independent from the initial cooling field value. Synchronous with the occurrence of positive exchange bias, distinct changes in the magnetization reversal process indicate a temperature-dependent rotation of the effective anisotropy and exchange bias axis. Model calculations based upon the electron microscopy-determined epitaxial Co/CoO-interface structure corroborate this interpretation.