If pumping is required then the mixture is probably lean and it's only the enrichment from acceleration that is keeping the engine running. As John said, suspect the temperature sensor or it's wiring. Maybe vacuum leaks that close up from expanding metal. It is possible, but unlikely, that the fuel pressure is too low when the fuel pump is cold. Sounds like the cold start circuitry is fine, it usually only functions for a few seconds and often only during cranking.