I did some looking to this after a question was asked about the source of the notion.
This whole idea of a cold moon seems to stem from Victorian flat Earth nutjob Samuel Rowbotham claiming that there was a Lancet article proving that concentrated moon rays cooled a thermometer. The March 8th of the Lancet from 1856 has this
https://babel.hathitrust.org/cgi/pt?...ew=1up;seq=267
which is about insanity and discusses the 'Morbid phenomena of lunar light'.
It does not report directly any such experiment, merely mentioning observations made elsewhere - some of which are unsourced hearsay. It also reports contradictions to those observations. The article refers to François Arago, an eminent French astronomer who did pioneering work on the nature of light.
In Popular Astronomy Volume 2, published in 1858, Arago refers to comments made by Louis XVIII to French scholar Laplace about the influence of 'red moons' on harvests. Arago went to discuss this with the king's Parisian gardeners and found them to have a number of observations about the moon on plants - observations mentioned in the Lancet article. Arago, however, found such observations to be unfounded and contradicted by evidence from experiments with thermometers.
This is the Popular Astronomy volume
https://books.google.co.uk/books?id=...me%202&f=false
and you can find the report on page 309.
Rowbotham's statement would therefore appear to be a either a misunderstanding or a deliberate misrepresentation of reports from other studies.