Only trouble with that idea--you will never get back home! That said---some have gotten many years of use from their Hydro-static tranny, even with hills, etc. And, others haven't! I know a man who didn't have any hills in his yard, just a strip about a hundred feet long, and his tractor lasted 10 years or so, and i bought it for parts when he sold it, as the tranny had started groaning, which is a good sign it was bad--which it was.
Way i figure-buy the machine--buy a good one--use it for what it was intended to be used for, and don't worry about it. Those tranny's are a "crap-shoot", meaning ya might get 10 years out of it--ya might not.

It also depends upon the quality of the hydro. My Wheelhorse hydro went bad after about 20 years and the new one has been doing well for about 10 years. I have some slopes of 20%.

To answer the part about "less strain" going down hill.
Yes, there will be less strain because gravity is giving you a boost instead of the engine and transmission having to fight to overcome gravity as when you go uphill.

Would it be worth the trouble to always mow down hill?
I really doubt it would make much difference in the long run.
Heat and worn out oil are really the killers of hydrostats when they are overworked and undercooled.
In theory, you would think maybe avoiding an uphill run would/could result in longer life for the trans, but if you are doubling the driving distance in order to always mow downhill, the extra driving distance probably cancels out any benefit you could get from always mowing down hill.

If you want to improve the life of the hydro, make sure the transmission case is always clean and free of grass or other debris that insulates and blocks cooling air from the fan.
Always run the engine at about 75% to 100% of the governed RPM. At low RPM, the fan is not turning fast, so it does not wash much air across the trans case.
At low RPM, the oil flow inside the trans pump and motor is not very energetic, and oil is not splashed/thrown/sprayed on the trans case as much as at high RPM, so more heat stays in the oil instead of being drawn into the case casting metal to be dissipated to the air from the fan.

Change the hydro oil at least every 3 years. Even if it requires removing the transmission assembly for an upside down drain, it is worth the effort.

Excellent advice Mownie very concise ! :)

Mownie, once again it is the M word . MAINTAINANCE, good job,also I will add keep debris off the tranny it will hold in heat , compressed air or a good leaf blower for that task.