I have this same problem! Although I did do some research, and I'd suggest messing around with EAC itself. Try installing it, reinstalling it and messing around with C++ redistributable installations. My setup is running an older 7th gen i7 and the AMD HD6870 which very well may be the issue. I don't plan to be able to run my game at a respectable Frame Rate, but if I cannot get it to launch, I may switch my card to my GTX 660. I'll tell y'all If that fixes it.
Tip: try unplugging all peripherals right after you launch it, even the mouse, as it has been peripheral related problems in the past.
IF YOU HAVE AN INTEGRATED OR MOTHERBOARD GPU, try turning off your pc, pulling out your GPU and changing to whatever your integrated is. If this goes according to my thinking, this should trigger Apex into giving you a GPU warning and may fix the problem (also may not lol), but it's worth a try.