I have a pajama set that I bought from Jos. A. Bank quite a while ago. I really like the crisp cotton during the summer. I’m not sure what makes them like that but I think it’s long-staple cotton. The label just says 100% cotton.
They don’t sell them anymore, at least not on their web site. I haven’t been into a store to ask in person.
Most cotton pajamas I see are very soft, loungewear, more like flannel. The category on their web site is “loungewear” with “pajamas” nowhere to be found. I like those but for winter. In summer I like something that is more like a dress shirt.
Has the style just changed so the loungewear type is just the popular one now? Or can I still find a nice set of crispy cotton?
This sort of thing has to do with the weave as much as the contents of the thread. It’s kind of like the difference between bedsheets that are percale (cool and crisp but not as smooth) vs. sateen (smooth but not crisp) vs. flannel (soft and fuzzy).
The trouble is, online this difference in material type isn’t really mentioned in a lot of men’s pajama products. But you can try with search terms like “light-weight” and “crisp.”
Of course, if you have a butler to iron them every day, linen is also nice for summer pajamas.
I have a couple of linen shirts but have never had linen pajamas. I have to iron the shirts or they have no body and look frumpy. My butler quit during covid. 
and no way am I ever going to iron pajamas.
If you’re looking for summer pajamas, you can’t top silk. I got pajamas from Lilysilk, custom-made by giving them measurements. They fit perfectly. Pricey, of course, but you wear them every day. And they come in a variety of styles.
The word you might be missing in your searches is “broadcloth”. Those are the pajamas that aren’t flannel.
Check the Vermont Country Store catalog—they have lots of men’s pajamas.