if it is browned up, it isn't palatable and won't be eaten. Could be kept up for ground protection, but there is no nutritional value in it. By ground protection I mean it is just a thermal barrier and weed barrier for the underlying dirt.
So, you can mow it anytime or you can leave it up until fall. Up to you. My guess is if you leave it stand it will become infested with weeds over the summer as animals and weather beat down the stand to ground level.