These transmissions aren't intended to be serviced, but there are some things you can do despite Tuff Torq's best attempts at preventing you from doing them.
The first thing to be sure of is that the drive belt isn't slipping. Be sure that the idler pulley/pulleys is giving good pressure on the belt and that the belt isn't glazed over.
After that I would check that the valves are operating properly. Get underneath the mower (not running) and have someone operate the controls. I couldn't figure out completely from the pictures of your mower how the controls work, but I assume there's a gear shifter and a engage/disengage shifter. Just look for a something that's not working as it should. Internally a valve may not be opening and closing properly, either from damage or carbon build-up, but you probably won't be able to tell that for sure. While you're under there, go ahead and clean the fan and as much of the transmission as you can get to. Heat is the real killer for hydrostats, keeping it clean is cheap insurance.
I don't believe they've made it easy to check the oil level in the transmission, so you'll have to be satisfied with making sure there are no leaks. There should also be a vent of some sort. It may be a rubber hose with a small screen or filter or it may be a small screen or filter stuck into the side or top of the transmission. Try and find this and see that it's clean.
Next step is to purge the system. This is fairly easy and I'll post the directions from Tuff Torq's site.
If all this checks out okay, then I think that you could use an oil change. I'm assuming yours is the one without the external oil reservoir? There's no plastic tank on the top of the transmission is there?
There's two ways to change the oil on this without opening the case, which I wouldn't do unless the oil change doesn't work. You can remove the transmission and turn it upside down and drain the oil out of the fill hole, or you can leave it in place and siphon it out. I prefer to siphon the oil out using an oil change siphon. I'm not familiar with how your mower is made, but on most mowers, there will be some sort of access hole under the seat, sometimes under the battery. Some you have to access the fill hole from under the mower. I don't work on Tuff Torq very often, but I think the fill plug is on the right side. Refill with 1.9 liters of 10W30 motor oil. You'll need to purge the system after changing the oil. For purging directions, see: http://www.tufftorqservices.com/isroot/tts/FlatHTML/TechInfo/docs/AirPurging.htm
This has about a 50/50 chance of fixing things. Other things that could be a problem are a stopped up internal filter, worn pistons on the pump, poor sealing in the pump, and excess carbon build-up. Rebuilding these things is somewhat difficult and you must have a manual to check clearance specifications. The internal filter may need changing, this too is on the fairly difficult level and a manual is pretty much essential.
I hope this has been of some help, please don't hesitate to ask if there's any questions. Thanks, PK.