Wof,
More brake energy is consumed (or created, as you will) stopping a heavy object, than a light one.
Brake energy isn't the issue with an overweight landing, however (though hot brakes can be); there are stresses on landing gear, wing spars, the fuselage, and many other components that are not what the airplane was designed to do. Add to that a hard landing, or a landing in excess of a given rate of descent, and these stresses can be magnified considerably.
Brake temperatures reflect the amount of braking that must be done to slow or stop the airplane.
A lightweight airplane with maximum braking may have hotter brakes than a heavy airplane that uses full reverse and rolls to the end. Where brakes are required, however, for a given stopping distance, the brakes will have to work harder to stop a heavier airplane, and consequently be hotter.
We frequently either land at our maximum landing weight (630,000 lbs), or we're landing nearly empty (on a ferry leg). There's typically a big difference in brake temperatures between a max weight landing and an empty one.
Ever arrive at a stoplight in your car when it's full of friends and heavily loaded, and notice a difference in the amount of braking you have to do to stop the car? It's no different than it is landing an airplane. A heavy airplane moving at a given speed has a lot more energy to stop, which translates to a heat increase in the brakes, than a light one.