Temperature sensor could be potential culprit. The temp gauge does run off the same sensor, but its resolution is not fine enough to see small temperature issues that could affect cold startup. But the temperature sensor generally gets worse at the temp extremes (ie. colder/hotter weather should make it worse - not a cold startup).
I agree - kind of weird that it only does this once a day, then be fine several hours later. Sounds like it needs a cold soak before the problem manifests itself. Sounds like an intermittent problems, which in itself is hard to diagnose. Problems could be anything from a vacuum leak to fuel issues.
I'd start with a good tuneup, if one hasn't already been done - just to start from a clean slate. Then go from there.
2002 Corolla S, Silverstream, 1.8L 1ZZ-FE (VVT-i)
2003 Matrix XRS, Cosmic Blue, 1.8L 2ZZ-GE (VVTL-i)