What kind of sling are you making?
Linen might be very nice for a ring or pouch sling. I'd pick a pattern/color that wouldn't show every little spot, though, kwim? And maybe have a spare sling, too, 'cause I think linen has to hang dry. Of course I think everybody needs a spare sling anyway

I had a wrap sling made of cotton gauze (aka "safari cotton" or "crinkle cotton") and it was really nice & cool. Wouldn't be durable enough stuff for a ring or pouch sling tho.
The style of sling definitely makes a difference in how cool it is too. I'd run, not walk away from anything with padding. Pouches don't have much extra fabric to feel smother-y. Asian-styles are barely there at all ... but I think they're not great for babies with little head control, am I right?