It may depend on the variety, but last year my broccoli produced a lot more side shoots as the weather cooled down in the fall, than it did in July. You just need to decide which you would prefer to have, more broccoli side shoots, or new plants. I am leaving mine in, and planting fall crops in other beds.

In my experience, the cut stems eventually develop pockets of rot and the weakened plants never recover good vigor. I'd take them out and start over with something new for fall. My brocs and close relatives came out three weeks ago and I was glad to see them go.

Thank you both! It's showing no signs of cooling off anytime soon (and I don't have another spot for fall crops), so I think I will pull them.