It could be a "gentleman's agreement" between the schools to cancel games if it's too cold. It might also be a special rule in a specific league. Either way, it's not an official high school or OHSAA rule.
The most miserable high school baseball I ever did, cold weather-wise, was about 35 degrees with gusting winds over 30 m.p.h. The wind chill was about 20 degrees. Had on long johns and thermal shirt, a couple of shirts over that, a jacket, gloves and it didn't make much difference. My nose, ears and fingers were going numb.
Had another game at Pickerington Central last year where a huge storm rolled in and it was blowing rain, literally, sideways. I was scanning the horizon for tornados. Thoroughly soaked and drenched me right through every layer of clothing I had on, like somebody had dipped me in a bathtub.
Couldn't figure out why the plate umpire wasn't putting teams in the dugout during this monsoon. You could barely see the ball, kids were slipping around, the temperature dropped and I was shivering. We played right through it- the worst rain I've ever played in. At the softball game on the next field, I could see them clear the field and everyone was heading for their cars.
After the game, the first thing I asked my partner was why in the heck he didn't get us in the dugout to wait out the rain. He said that he had worked a game for Pickerington a few days before that he suspended due to rain, and the Pickerington coach some grief about that. So, he was determined to make him play through the downpour in this game as a payback!