no the lid should face up and keep them at room temp!
My room temp is 73-74 degrees, that is what we keep our AC on pretty much everyday year round. They grow like weeds at that temp for me!
I thought refrigeration kills hornworms? Also if they are kept at room temperature they grow rather slow. The last pod of them I had took over 3 weeks to reach +4 inches while other pods took less than a week when it was warmer in my house. I wonder how fast they grow in the summer....