The broccoli (I think it's brassica family) do prefer much cooler to freezing temperatures. along with cauliflower and cabbage, brussel sprouts, etc. they will bolt (throw out flowers) very quickly even in mild heat

Once you cut the central head ( regular not sprouting broccoli) you will get side shoots but they get smaller with each cutting. When the sprouts get too small and ragged for your use, its time to abandon the plant. They do tend to get a stronger flavor as the temps rise. Both the main head and the sideshoots are flower buds, the trick is to use them before the flowers open.