As we know, our Moon is a lazy spinner. It takes about a month for a full circle around its axis.
It means, that the lunar day and night are each as long as two Earth weeks.
Every Moon meridian enjoys quite a long high noon, consequently the rocks, pebbles and dust on the surface warm up to about 120 degrees Celsius, a little while after the Moon noon.
The surface will have about fourteen days to cool down, during the long lunar night. It gets really cold, 200 or 220 degrees Celsius below zero.
Regardless, the Moon heat machine has very little choice what to do. It absorbs and emits all the Sun’s energy it gets, just relaying the power of P Watts, converting mostly short EM rays to mostly longer EM rays and that’s that. The Moon is just a short stop for the photons traveling from the hot Sun to the cold space, there can be no change in the amount of energy per unit of time.
Let’s say that the Moon completely ceased rotating. It will still transmit the same power P from the Sun to space.
Then let’s say that the Moon started rotating as crazily as the Earth. Still it will transmit exactly the same power P from the Sun to space. The Moon has no say in this.
But the afternoon’s hottest rocks on the surface of the now rapidly rotating Moon will not be as hot as they are now. They will be a bit colder. And because of the Stefan-Boltzmann law, their output in the form of IR radiation will also decrease. One percent colder on the Kelvin’s scale, means about four percent less radiation.
The former hottest rocks therefore now contribute less to the whole P Watts strong output. Disproportionally less, so something else has to make up for it. There is only one way. Those colder rocks must be hotter now. How much of them and how much hotter? The same surface of formally colder rocks, now warmer for the same number of Kelvins, isn’t enough, as the output increases with T^4. So only a bigger area can compensate.
The average surface temperature went up this way, just to abide by the energy conservation law! A faster rotating planet, everything else being equal, is warmer!
A long time ago the Earth was warmer, despite the fainter Sun. This is called The Faint Young Sun Paradox.
There are no paradoxes in real life. There is no paradox here. Paradoxes only live and prosper inside mortally wounded or ill theories. Our planet was warmer then, due to its faster rotation. Now the Earth is cooling in the long run, because of the decrease in its rotational velocity, caused by tidal forces.
Every planetary scientist should know this. Rotating Bbq Grill operators have always known this fact.