What would cool it off? Even though space is cold, computers are normally cooled by contact with other particles (conduction). That's why a heat sink works better than air, which works better than nothing.
You would get some radiation heat loss. The chip would heat up and give off infrared radiation, which would cause some temperature lowering, but it would be pretty subtle until the temperature got very high, and the amount of heat lost through radiation isn't really affected by ambient temperature, only by the temperature of the radiating object.
Keep in mind that temperature is determined by how many air molecules hit an object and at what speed they hit (the kinetic energy in the air). So if you have a vacuum, there are no particles there to hit it, so it doesn't really have a temperature (in the vacuum itself). If you take a hot object and throw it out in space in a complete vacuum, where it's not getting any EMR, it's just going to slowly approach absolute zero in temperature as it radiates away heat.