It’s very common for horses to want to rush up hills because it takes less work on their part than walking up the hill at a steady pace. Once a horse has rushed uphill a few times, it starts to become a habit.
Find a hill to practice on that isn’t very steep. It can be tricky doing a lot of starting and stopping on a steep hill, so that’s why it’s nice to practice on a hill that isn’t very steep in the beginning. I would recommend riding part way up the hill and stopping your horse, waiting a moment, and then going a little farther up the hill, then stopping again. Keep repeating this until you get to the top of the hill. Don’t wait for your horse to start rushing before you stop him. If he’s traveling at a nice slow place, which is what you want, stop him anyway to really reinforce the idea that he needs to travel slow. However, if he starts to pick up speed, definitely stop him at that moment so he knows that wasn’t what you wanted him to do. The stopping on the hill keeps things slow, and the rushing will start to go away.
Once you get to the top of the hill, I would recommend riding back down the hill and repeating this exercise a few times. Keep in mind that the horse thinks rushing to the top of the hill is the best way to get out of work. However, if you go back down the hill and repeat the exercise a few times, he will start to realize that rushing didn’t get him out of the work after all.
Repeat this on different hills if possible. The more your horse wants to rush, the more times you will have to stop him for the exercise to work. As your horse gets better at this and loses the desire to rush, you can decrease the number of times you stop on the way up the hill. After a while you won’t have to stop at all.
It’s very important that you don’t let your horse rush uphill sometimes and then try to stop him or slow him down other times. You have to be consistent and never allow rushing. If your horse is allowed to rush some of the time he’ll likely want to rush all of the time.
Something else you can do is give the horse some work to do once you get to the top of the hill. This can be anything that takes some physical effort on the horse’s part – trot circles, several turns, etc. This also helps get the point across to your horse that rushing to the top of the hill didn’t get him out of work – instead there’s actually more work up there, so he won’t be as inclined to hurry to the top of the hill.