If my understanding is correct, the temperature of space (as defined by the temperature that a black-body will reach) has been decreasing since the big bang. It has never increased. Additionally, because of the existence of dark-energy, it seems most likely that the temperature never will increase.
Taking the large-scale universe as homogenous, how many times greater would the energy release rate of all cosmic processes have to be in order for the average temperature to hold constant against the cooling effect of dark-energy?