Actually it makes it harder. Without air to use as a heat transfer medium you don't have anything to bleed heat into.

On the contrary, it makes it much harder. On earth, you can radiate away heat by direct contact with air. No air in space means the only way to dump heat is by black body radiation.

It's a common misconception that space is "cold". This is due to the definition of "cold" -> the kinetic energy of particles in certain volume is low. But while on Earth is means you have lots of particles, each with low energy, in space it's very different - you have very few particles, often with very high energy.

In order to cool something down, you need to transfer the energy. On Earth particles with low energy will steal some of the energy of your hot thing, cooling it down. The more particles, the better. In space this effect doesn't exist, there are no particles to steal the energy. You need to radiate the heat as infra-red.

Heat can be transferred in three ways: Radiation, Conduction and Convection. Conduction is heat transfer by two solids touching. Or when you put your hand on your heater, you feel your hand getting warmer. Convection is heat transfer to fluids (gases). That is why the air in your room heats up when you put a heater in it. Radiation is the weakest form of heat transfer, by far. But if you aren't surrounded by an atmosphere or ocean to dump heat into, and instead are surrounded by a vacuum, radiation is the only way to get rid of heat. Conductive transfer will constantly heat up the spacecraft until parts start to melt if you cannot radiate the heat fast enough. Hence the need for large radiators on spacecraft that produce a lot of heat, e.g. the ISS or even the shuttle, whose whole payload bay doors were used as radiators and needed to be opened somewhat quickly once in orbit or the shuttle would overheat.