Would vacuum be lower when cold?
yes, due to the higher revs/slightly more open throttle/IACV w'ever. vacuum would be reduced.
That's why I wondered if, say, first thing on a cold morning under conditions which you feel the car would exhibit poor initial braking (*breath!*) you instead, allow the car to warm-up for 15 minutes before moving it. The revs should have come back down and the engine should be warm at that point and have had an opportunity to build vacuum.
Then monitor the performance when you pull away. If it fails, the problem is less likely to be the booster because, vacuum should have had plenty of time to build. If braking is normal, the problem is more likely to be the booster.
I dunno, maybe I'm not thinking right about this.
The booster, with a functioning check valve, 'should' hold a vacuum for a significant amount of time. It's designed to provide 1 or2 'boosted' brake applications should the engine die on the road. I suppose you could pull the hose off it in the morning to see if you hear air rush in or not.