I've connected my apple magic keyboard to my phone before, mostly just to see if it worked. It's a small compact thin and light keyboard. I'd imagine most keyboards with bluetooth these days probably connect, just depends on what kind of form factor you're looking for
I bought one of these in 2018 on ebay for my dad, and had it shipped from japan.
Living in the US I had to jump through the hoop of making a fake japanese icloud account just to be able to access the app on the app store, which is how you control the device.
Ultimately it seemed like a really cool device but at least at the time wasn't very effective or practical for cooling down, at least in my and his experience. This was mostly just being outside, doing yard work etc in the hot summer.
I think buying one of the <$50 portable neck coolers on amazon are probably just as if not more effective