space is cold.. very cold...
It might be cold but first and foremost it's vacuum.
Being vacuum, it only lets you get rid of heat by radiation.
Ever used vacuum flask?
It will not heat up becasue of Stephan-Boltzmann law. The heat would be radiated away rapidly.
Radiator efficiency may scale with 4th power of its temperature, but there is this pesky constant with 10^-8 which happens to matter at relatively low temperatures:
1. You'll generally want your radiator below 3000 something K, realistically probably below 1000K. Wolfram melts at 3700, carbon at 3800, most materials can't handle that much and most materials lose structural strength when heated sufficiently. You will also want to keep some liquid coolant in and circulating. At 1000K you get some 50kW per square m if your radiator is close to black body.
2. Power requirements of sensible space drive are hilarious. You need to make as little propellant as possible carry as much momentum as possible (to not run out of remass). To do this you must fling it really fucking fast. Unfortunately, kinetic energy grows with second power of velocity while momentum only with first, so your energy requirement will go up pretty fast and so will amounts of heat to radiate.
Now, depending on drive, propellant may help carry the heat away, so it may not be as much of a problem when flying, but if you want combat in your game and weapons in form of railguns (that nom power) and lasers then you'll have problem unless you just boil your coolant into space, but that will end pretty quickly, along with coolant (or remass if you use it instead of your circulating coolant for lossy cooling, which makes sense as your remass will likely be hydrogen, as this shit is all over the place in space).