Opposite for me with the Note 4 and correlates to reviews by Anantech and GSMarena (great Note 4 battery life).
I find for my medium to heavy use the Note 4 is significantly better in regards to battery life and heat issues than the Pure. 3D games and HD video runs ten degrees warmer and screen on time is three hours less on average with the Pure. No rational comparison. The heat and sharp battery decline with GPU heavy apps seems unusual with the Pure and Moto needs to manage the governor settings better with the GPU (if they can).
If I just used Dolphin and Firefox, streamed video (not HD), or used lighter GPU apps, the battery life was decent and heat relativity normal with the Pure. Ditto for game emulators since they are very light on the GPU and very heavy with the CPU. After ten days now with the Pure, it seems the Pure is a great device for light to medium users that do not use many apps GPU heavy. The GPU seems the problem on this device.
Per Best Buy, they are getting returns due to the battery and related heat issues. I do not think it as much a product defect as it is the 808's gpu is having a hard time with 1440p and needs to be throttled. That might be the catch- if Moto throttles, the GPU might have a harder time with 1440p. This seems not a lot different than the G4 issues.
Like the G4, most people are light to medium users, so usually no harm no foul for them