There are two parts to this book's statement:
1) That nights with a full moon are colder [than those without]
2) That cold temperature increases hydrogen peroxide formation in dew.
For the first part, this stack exchange page addresses the issue of the moon's affect on the weather (summary: negligible):
This page shows the high and low temperatures in August 2015 in the city I live in.
Full moon was on August 29 when the low temperature was 73 degrees F. This was the 2nd HIGHEST low temperature for the month.
This page from the U.S.National Weather Service shows high and low temperatures in the city of Binghamton, NY for every day in 2015:
This NASA.com page gives the dates of full moons in 2015:
Here is a summary of a comparison:
- Jan: Night of the full moon was the HIGHEST low
- Feb: 9 nights with LOWER temps
- Mar: There was only 1 night with a LOWER temp
- Apr: 3 nights with a lower temps
- May: 7 nights with lower temps
- June: 2 nighs with lower temps
- July: Two full moons in this month (1st and
31st). 17 nights were colder than the coldest full moon night.
- August: 3 nights were colder
- Sept: 3 nights were colder
- Oct: 3 nights were colder
- Nov:3 nights were colder
- Dec. 18 nights were colder
There are ways (other than monthly) that the data can be grouped. Here is a summary of the 7 days before and after each full moon:
- Jan 4: 71% were colder than full moon
- Feb. 3: 14% were colder than full moon
- Mar 5: 57% were colder than full moon
- Apr 4: 21% were colder than full moon
- May 3: 35% were colder than full moon
- June 2: 14% were colder than full moon
- July 1: 78% were colder than full moon
- July 31: 85% were colder than full moon
- Aug 29: 14% were colder than full moon
- Sep 27: 50% were colder than full moon
- Oct 27: 0% were colder than full moon
- Nov 25: 21% were colder than full moon
- Dec 25: 64% were colder than full moon
I didn't run a statistical analysis to look for the amount of correlation between phase of the moon and temperature, but as a absolute rule (full moon is always colder [than nights without full moon] there are numerous counter-examples to show that is false. As a general rule (that may have exceptions) . . .well, there are LOTS of exceptions.
For the 2nd claim, that the amount of hydrogen peroxide produced in dew increases with decreasing temperature: Even if it is true, you would need to evaluate the risk to your laundry using exactly that: Temperature (not phase of the moon).