This could make an interesting experiment for a high school or middle school science project.
"Hypothesis: sunlight warms and moonlight cools."
You'd need a relatively open space (yard or field) with a fairly uniform surface where the experiment wouldn't be disturbed, in an area free from reflections of moonlight and sunlight from windows or other surfaces, a way to support an obstruction, say a square of thin plywood maybe a half a meter to a meter on a side, horizontally about 4 times the length of a side above the ground to cast a shadow, and a way to record the temperature in at least two places, one that gets shaded from moonlight as the moon moves across the sky, and one that doesn't. Placing fast-reading temperature probes an equal distance from directly below the obstruction, so it blocks the same amount of sky for both would be ideal, or, perhaps a thermal imager would work if one was available. If the obstruction is 4 times as high as it is wide, the sensor will be in the moon's shadow for about an hour if the moon is near the meridian. You'd probably want to record the temperatures continuously for at least a half hour or longer before until that long after the shadow passes over the sensor.
Conducting the experiment during the day, with the sun's shadow passing over one sensor but not the other tests the "sunlight warms" part of hypothesis. It also tests the equipment and technique, so doing that first would probably be worthwhile.
Subtracting the temperature recorded from the control sensor (the one never in shadow) from the temperature recorded from the sensor that the shadow passes over should remove warming or cooling trends from the recorded temperatures, making it easier to see the effect of the shadow on temperature.
If someone has schoolkids and is looking for an experiment for them to do, this could be a good one. The point of a school science project is about designing and conducting a good experiment, with adequate controls, to test an hypothesis, and then analyzing the results and drawing a conclusion. The point is not whether the hypothesis seems preposterous or not, nor whether the result confirms or refutes the hypothesis. If it turns out that the experiment is not able to detect the expected phenomenon, examining why it failed and proposing an improved experiment is also a useful exercise.