I owned a new 34T from 2005 to 2009, so am reasonably familiar with it. The boat is fairly vice free. There are only a few common things that I can think of. One is the ZF Microcommander electronic throttle and shift. One some boats, ours included, the throttle and shift would suddenly go to neutral and idle. Cycling power restored normal operation. Mainship issued a service bulletin that suggested replacing a power supply relay located in the electrical chase. Some said it worked, others said it didn't and one owner was so frustrated that he replaced the whole system with mechanical cables. Another solution is the Kobelt equivalent electronic system that is reportedly rock solid.
Ask the owner if he has experienced any of this behavior.
You are unlikely to notice this in a sea trial as it is very intermittent. Just be aware and if it bothers you try replacing the relay. The service bulletin can be found if you root around here- https://web.archive.org/web/20070814...com/index.html
The other issue is that due to its rather wide beam and therefore aft transom width, the boat gets pushed around badly in a 3-4' following sea. With the standard autopilot it would do 30-40 degree S turns. You could hold it fairly straight by hand, but you had to work at it.
A fairly good solution was to add the gyro feature to the standard Raymarine autopilot. That quieted down the S turns to about 10-15 degrees. Most new autopilots have this feature standard and your 2008 might have it already.
It is a very nice boat for a couple as it just has the forward queen berth. Some couches pull out to make another bed but that means your guests are sleeping in the main salon. The fly bridge is huge and a great place for a cocktail party.
The single Yanmar 370 will push it to about 12 kts while burning 11-12 gph without stressing it too much. But we mostly cruised at 7 kts and burned about 3 gph. The engine room is open and it is easy to get around the single to work on all sides.
All boats develop their specific issues over time, but I can't think of any thing else common to all.