Question

Would you be doing more work going up the stairs twice as fast?

Answer 1

If you use the stairs at normal speed, you will survive longer, and go up higher.
When going up the stairs twice as fast, you lose energy more quickly, but height would remain the same.

Answer 2

Yes you would because if you go at a slower rate you don't feel the work that you're doing but if you go fast then you can tell that you put work to it and you can tell but how fast your heart is beating