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Really? After a road bike has already slowed to the MTB speed, it still has as much kinetic energy as the MTB, along with it's lower rolling resistance and less drag. Once the speeds are equalised, the road bike's brakes still have more to do than the MTB's

In other words, if you rode an MTB down a typical Alpine road descent, its brakes would have an easier time than a road bike's on the same descent.

Dammit asked about energy turned into heat, and i gave a slightly trolling answer. The key bits were assuming negligible pedalling, and 'on average'. The average is taken over the whole descent, including the bits where the riders aren't braking...

In more detail:
mass - the MTB is heavier, but both bikes are light compared to their rider, so the difference in total mass is small.
energy - i'm assuming no pedalling, just looking at the gravitational potential energy, so with a bit more mass the MTB has a bit more energy to get rid of, but it's close.
Rolling resistance - the sum of all the non-braking constant forces that resist motion. The MTB has more, but the work done against it is resistance force x distance travelled against it. With the road bike's much longer path length of descent, it's not clear which will loose more energy this way.
Drag force - air resistance and any other forces that go roughly with the square of speed. To calculate the work done against it you can integrate it over the path length of the descent. It will be much larger for the road bike because even though it's more aero, it goes much faster and further.

Brake heat energy - what's left of the starting potential energy after the above have been accounted for. It's clear the MTB will have more.

You're answering a more relevant but slightly different question about how hard the brakes have to work.