That doesn't sound right to me. If there's no air, then only black body radiation can be used to cool the data center. That means a massive radiator, a lot larger than a heat-to-air radiator+fan used on earth.
But geothermal cooling would be great on the moon too. Run a pipe 2 meters under the lunar surface and it is -21C.
I think the whole idea though is to make a low wattage space-stead so you can store copies of Moana out of reach of Disney cease and desist letters.
Isn't the moon geologically dead though - no water or geological movements?
I worry this would just result in the ground absorbing the waste heat and eventually becoming too warm to effectively cool anything. Especially because the ground itself would eventually still be limited by the rate of radiative cooling into space, right?
The problem isn't so much geological activity or lack thereof, as the nature of lunar regolith. Lunar regolith has a conductivity of 0.004W/mK. That is lower than aerogel! So unless the subsurface has a much higher conductivity, using subsurface cooling would be doomed.
Edit: Lunar Regolith is only the first 4-5 meters of the lunar surface.