You have two objects of the same temperature but differing sizes, which one gives off more energy?
Depends; are they also the same material? If yes, they give off heat at the same rate, but the smaller one will reach temperature equilibrium first.
I can't be sure from the info. Black body radiation will be the same per unit area, so I suppose the black body radiation in total will be greater from the larger object, assuming that it has a larger effective surface area, which is probable but not declared (same energy flux per unit area times larger area = larger total energy flux). A thin, flat disc that is "smaller" (less volume, presumably) could have a larger surface area than a sphere that is "larger" (greater volume) so the disc would have a higher emission flux. Does depend on what we mean by "larger" and "smaller". Also depends on what we mean by "gives off" energy. Normally, we do not mean conduction or convection when discussing "gives off", as that normally means "emission" (radiative heat, e-m radiation or "light").