The appeal of common IR photography is not so much that it reveals IR light, but that IR light gets reflected in different amounts than visible light. That's where the black sky and the white trees come from.
If you look at the moon, it's homogeneous, all rock and dust. It is unlikely that an IR image of that surface will look much different from what we see in the visible light range.
I'm not claiming an IR photo of the moon is interesting, merely that it being cold is irrelevant to getting an exposure from it.