I have tested it and noticed the device gets very warm at the display area and top during gaming. Battery life is an issue for all devices when 3D gaming, but the Qhd drains even more battery. To offset this there is throttling of both the display and the chipset. That said, the S4 and most other devices get very warm too. I mainly noticed the throttling with emulators, since they are VERY cpu intensive, so any throttling is noticed. 3d MAME games, PPSSPP and N64 are examples.
I am sure there are rooted methods to disable the throttling, but LG put it there for a reason. It will be interesting to see how turning it off pans out for the device.
The one device that seems the best for gaming IMO is the Note 3. It does not get as warm, but is a two hander for most people. I wish I held out several months and bought the Note 3 instead of the S4.
BTW, you can see the throttling when running concurrent benchmark tests. The first ones have higher scores than the following ones. One app test (can not remember which) though is ALL over the place for some reason and has no directional trend.
After all said though, probably best to test yourself

I can say if gaming is a priority and you do not mind a two hander, the Note 3 is hands down the best overall device for gaming and video media. Browsing too.
Added: I am close to talking myself into the Note 3, since I am concerned the Note 4 will have similar issues like the G3 as far as heat and battery life....