I feed freshly cut grass to my goats all the time when available. HOWEVER - not cut with a lawnmower, but the long stuff cut with the blade/brush cutter attachment on the strimmer. It stays long, contains quite a few thistles and other weeds and the goats go crazy for it. I get mine started on it slowly though, a little to begin with and only after they had a good feed of hay.
It often gets cut in several bags and fed over a period of two to three days. Yesit starts to get warm, but as long as the goats are used to it - mine have never had a problem.
At Great Yorkshire show the sheep/goat and I presume horse people get a twice daily fresh delivery of freshly cut grass - brilliant.