You can do more intense or challenging versions of your jump to get your heart rate higher.
Alternate 1 foot jumps, left, right
Alternate, 2 left, 2 right
Alternate feet, high knees
Or you can just jump faster.
Do some set time, or number of jumps. Use whatever jumping style you need to get your heart rate up, higher than you would jogging. Then take a brief rest, and repeat. Do some set number of rounds. Like intervals, or HIIT high intensity interval training. Like ZEB said, this is a popular way to train these days. You can gradually progress by doing more or longer rounds. Or doing the same work, but with a harder jumping style. Or decrease the rest between rounds.
So I guess theoretically you could do a more intense workout, in less time, with less pounding than jogging.
I don't mean to try to talk you out of running or anything. But jumping rope is cool. Even just as a change of pace, or something to do when it's raining.