In deep craters near the Moon's poles, permanent shadows keep the surface even colder — NASA's Lunar Reconnaissance Orbiter has measured temperatures lower than -410°F (-246°C)
That's not the issue, though. In a vacuum there is no medium with which to carry the heat away. You can't send it into the air with fans or heat sinks because there isn't any air.
At least on the moon you could sink it into the ground. But in orbit you don't have that luxury. This is a major problem that spacecraft and satellite designs need to work around, and much effort is expended in that department.
Even though space is generally considered "cold," in the absence of a medium to sink heat into the best you can do is rely on infrared radiation which is not terribly effective.
What would they use it for? The 2.5 seconds of latency would be too high for most uses. Cooling will be very difficult with no atmosphere. Solar power will be hard since night time lasts two weeks. Radiation will damage electronics unless they bury them.
Seems like the moon would be close enough for our standard IPv6 TTLs to work, but it seems more likely that we will have to abandon domain names in favor of something like IPFS, since it's a resource locator instead of a location locator. If you were on Mars, for example, you would not want to have to contact Earth every single time you wanted to load a web page. And so you would contact Earth the first time to load it. And then it would be saved locally. And so anybody who requested that page in the future would talk to you instead of Earth.