It's probably just fall decline, but, it could also be a fungus called rust. It's easily treated with a fungicide, but it is a good idea to remove the infected leaves.

Its where the leaves were damaged and the edges are "rusting"... your hosta contains iron... iron oxide forms where the iron is exposed to the elements... Its too late in the year to worry about fungus....or fungicide... thats for the growing season... I agree that you should treat with a fungicide, but wait until spring, then treat all season according to directions... but I would just leave it alone til you have to do fall clean up. Remove all dead hosta foliage, atleast, from your plantings. If you are worried about slugs and fungus, you should remove all old mulch, leaves and debris. Then remulch with new mulch.
Hope this helps,
John