Well, my guess is that people run it at 0C as a first try. If it works at 0C, I would also keep it there.
I know that some "activated ester" analogues are somewhat thermally unstable and that's why I wouldn't heat it up unless needed.
E.g some mixed anhydrides from isobutyl chloroformiate have a survival of just an hour or two, even at 0-5C.
It would be interesting to know if somebody has a rough idea about the thermal stability of EDC/HOBt-intermediates.