Here’s the plot of the speeds vs. distance for the three cases. 20 m/s is about 45 mph, and 55 m/s is about 123 mph. The slow braking was done at 0.4G with the hard braking at 0.8G. Acceleration for all three cases was 0.4G. On Case 3, it goes up to 55 m/s like the other two cases, but then drops to the speed required to match the time of Case 1; not completely realistic, but it was simple.
Here you can see that Case 2 hits a higher peak temperature than Case 1, but also cools down to a lower temperature. Case 2 gets cooler because the car spends more time at speed and less time on the brakes. Case 1 has a lower peak temperature because the brakes reject heat while braking; as the car spends more time on the brakes, it rejects that high heat over a longer period of time. Case 3 stays the coolest spending the most time off the brakes. 700K is about 800F. 620K is about 656F.
Perhaps a more telling graph is comparing brake rotor temps over time. Looking at the areas under the curves, Cases 1 and 2 look about the same. However, Case 2 completes my hypothetical circuit much faster. Case 3 stays significantly cooler than the other two.
The lesson here is that harder braking is better. Either you end up going faster and have basically the same rotor temps, or you go the same speed with lower rotor temps as compared to being too light on the brakes. Lower rotor temps should translate into less brake fade and also lower pad and rotor wear. Harder braking means less time on the brakes too, so that may also translate into less pad and rotor wear. So don’t be afraid of the brake pedal and stomp on it!