So I will disagree a bit with RD. In my opinion, and you should take it with a grain of salt, I would switch my Hydraulic to 0-40 synthetic if I lived in a cold weather environment (I do not, freezing weather is only 5 days a year and only just freezing in my area). I would also change my engine oil out to the same as well.
The first number, the 0W or 10W is a number that represents viscosity in cold weather. If you live in Pheonix, you run 20 -50 as it holds up better to high temp. So on your machine, the last number, the 40, is what is important to the system, it is the viscosity when the oil is hot. The zero says the oil is pretty fluid when it is cold (not thickening up).
I feel legally that as long as you are putting in the correct top end numberd oil, you are not breaking the warranty.
PT is fantastic because it is a relatively simple machine. You look at GEHL or some of the other slope machines and they are fancy as all heck, but there is a lot more to fail. The engine on a 425 is basically a lawn mower engine, IMO it was never designed to turn over a pump in the cold. You find this engine on mowers that are not engaged to anything until you tell them to be. PT should have probably put a clutch on the system, but it would only be another point to break, and add a myriad of design issues that would escalate the price. What I am getting at is that the thinner cold oil would be easier on the system, and no reason a zero weight would be an issue.
As for Synthetic, I personally feel (and have nothing to corroborate this with) that the oil runs thinner, has less friction, which will help in cold weather starting as well as keeping your engine cooler.
As for brands, people say Rotella is the best, people say Mobile One. I tink at the end of the day you have to do some serious research (there are good places on the net to do this) but that there are very good synthetics that cost much less than the Rotella Mobile one and do the same job.
For my machine, it leaks oil like it is going out of style. My engine I run Mobile One. My Hydraulice I use the cheapest NAPA / Walmart / Costco oil I can find as it tends to get sprayed all over the woods in short order.
Motor Oil Viscosity Grades Explained in Layman's Terms