ISRO Chandrayaan 3 released the following data today about the temperature characteristics of lunar soil.
Clarification: I do not claim it to be correct usage but here is what I meant: Primary conclusions are those that are directly visible from the data. Secondary conclusions build on the primary ones, and so on.
Looking at it with minimal prior knowledge, I think you expect that behaviour if you are on a surface that is sunlit (or recently sunlit) on a body that has no atmosphere.
The very top surface is hot because it absorbs sunlight. It can achieve an equilibrium temperature by radiating this heat back into space or conducting it below the surface. Since there is no atmosphere, then cooling via convection (as would happen on the Earth is not an option).
If the thermal conductivity is low and the top surface receives of order 1 kW/m$^2$ (assuming a low albedo), then the surface radiates at a temperature given by $$\sigma T^4 \simeq 1000\ ,$$, where $\sigma$ is the Stefan-Boltzmann constant equal to $5.7\times 10^{-8}$ in SI units. This would give $T=364$ K (91C), which will be an upper limit because some light is reflected and since Chandrayaan-3 is at the lunar south pole, the sunlight comes in at an angle that probably reduces the flux from my estimate.
If the thermal conductivity is low, and there is no atmosphere or other fluid to transfer the heat downwards, then the temperature will drop rapidly with depth until it reaches the equilibrium temperature of the Moon as a whole. This will be a rough baseline temperature that could be estimated by assuming the Moon as a whole must on average radiate all the sunlight absorbed. Again, ignoring albedo, $$4\pi R^2 \sigma T^4 = 1200 \pi R^2\ ,$$ where $R$ is the lunar radius (which cancels) and 1200 W/m$^2$ is the flux of sunlight incident normally on an object at the distance of the Moon from the Sun, with roughly 10% reflected back into space (the lunar albedo). This gives $T\simeq 269$ K (-4C). This is probably an overestimate if heat transport within the Moon is ineffective, since the lunar south pole will receive less than its fair share of this flux.
Thus I would expect to see a sunlit temperature profile as in the picture, with a high temperature at the surface reducing to minus a few Celsius at depth, but for that trend to reverse when the surface is not illuminated. i.e. For the surface to be well below zero Celsius and colder than the interior, which will remain at minus a few Celsius. That is because the surface can radiate more efficiently than heat can be brought to the surface by thermal conduction.
The surface gradients on Earth will generally not be as sharp because Earth has an atmosphere and other fluids that are able to transport heat much more effectively over these short distances and to cool the surface. The equilibrium temperature of the Earth (i.e. the baseline temperature) is also higher than the Moon because of its atmosphere. I would guess that the closest you would get to lunar conditions would be in a desert, where it is very dry.
