Which emits more radiation - a warming blackbody or a cooling blackbody? My intuition says a cooling blackbody would emit more radiation as any body that emits radiation is itself losing that energy, causing its internal temperature to decrease.
 A: The amount of radiation being emitted by a blackbody doesn't depend on the time-derivative of its temperature--so, at a given temperature, otherwise identical blackbodies emit radiation at the same rate regardless of whether their temperature is increasing or decreasing. It should be pointed out that the process of blackbody radiation always contributes towards decreasing the temperature of the body as it contributes towards losing heat (and except for black holes, everything has a positive heat capacity). The reason behind the overall increase in the temperature of a blackbody (if and when it is happening) would always be external--for instance, it might be being incident on with some external radiation at a higher rate than the rate at which the blackbody is emitting radiation. But such an external process plays no role in determining the instantaneous rate of radiation/heat emission by the blackbody--at each instance, this solely depends on its temperature, not on the time-derivative of the temperature.  
Edit
I realized the physical intuition that might be leading you to ask this question so I would add a few words here. I think you might be thinking of a process where a blackbody is being rapidly cooled down and thus it would release a large amount of heat in a short time. This might have led you to think that since the blackbody is releasing a large amount of heat in a short time, it should be emitting out more radiation. This breaks down because, as I mentioned, at a given temperature, a blackbody radiates at a fixed rate which is determined purely via its temperature (and geometric dimensions)--so each blackbody at a given temperature is always cooling itself via the process of radiation emission at the exact same rate. Now, if an external mechanism is cooling the body additionally, then certainly the blackbody will give out more heat at a faster rate--but this increase in the rate at which it gives out heat will not be in the form of radiation emission--it will be via other forms of heat emission based on how the external mechanism is interacting with it. For example, if a blackbody is being quenched in a bowl of cold water, its rapid loss of heat will be via the conduction mechanism between water and the blackbody--not via additional emission of radiation.
