> You seem to not even understand the very basics of heat transfer.
Basic physics: The moon is very cold in surface shadows and below the surface. It is an enormous pre-chilled heat sink.
The surface is also the support structure for any scale of radiative cooling with the same heat physics as orbit, but much better for larger and enhanced radiative engineering.
For example, heat pumps can centralize waste heat energy. Higher heat density vastly increases radiative efficiency.
• Permanent shadow: 40-60 ˚K, -230 to 210 ˚C
• In polar shadow: 25-30 ˚K, -250 to -245 ˚C
• Under 1 meter of surface, equatorial: 250 ˚K, -23 ˚C
• Under 1 meter of surface, polar: 200-220 ˚K, -75 to -50 ˚C
Many advantages beyond unlimited heat sink/radiative area: all compute in one place, i.e no size limit, so low inter-center latencies, no orbit safety negotiations or periodic orbit re-lifts required, able to update entire data center in a single trip, easier maintenance and stability in gravity on a surface, solar panels can be distributed over distance limiting total space debris risk, different component lifetimes don't result in wasted components, ...
Only downsides are a higher Earth-Datacenter latency, lunar dust resistant design, and a need to be at a pole for all-month solar power.
Nuclear power, or nuclear + solar, would allow any site.
Note that shade can be created anywhere on the surface via reflective shielding, and power can be used to heat, in order to stabilize temperatures in a desired band. Buried installations can use insulation for even greater temperature control.