If motor has high mileage (100-125K+) you can easily lose 3-4 inches of vacuum and run normally. Loss of ring and valve seal.
Look at O2 sensor output, if not switching right it will back up vacuum leak theory. Like said before anything that affects the closed throttle air flow there will affect cold start running, plugged pcv valve, IAC or TB aircrack. I often fix the cold start die with TB aircrack cleaning. May be like zetecs and show a hot idle issue from vacuum leak faster than at cold start, at least every one I've ever fixed of vacuum leak did that. The cold fuel added is more than the hot at closed loop and hot running will show idle issues before cold does. The hot idle trim hits limit before the cold does.
If anyone has messed with the absolute TB aircrack setting to throw off the IAC default running position, well, that can do it too. IAC takes a second or so to stabilize, if aircrack too small the motor will die in that. Older engines do not pull vacuum as hard, I correct cold idle dying on higher mileage car by slightly increasing the TB air by maybe 1/3-1/2 turn more to increase the amount a miniscule amount. You commonly end up doing that on oldschool non-PCM cars as well to do same thing. The loss of engine seal again, they don't last forever. Less sealed engines need more idle crack because vacuum has dropped, pretty much all engines do that.