Rob, I do leave the Jump Starter on the boat nearly all the time. The only time I take it home is when the indicator shows a low charge or once a month. It seldom shows a low charge so the once a month plug in is what I usually do.
I don't have cold weather issues but I do have intense heat in the summer. So far the jump starter has not been problematic. I do have a silver tarp rigged like a tent that covers the boat. In Florida the reflective tarp material is really the only way to go. The temp under the tarp really doesn't get any warmer than the ambient air temp. Other covers have not been as kind to the boat with 105+ readings under the cover. I think high temps can be as hard on a battery as cold weather.