Not likely the oil pump
or you'd never have oil pressure. Also probably not the oil pressure relief valve. While the valve is open when the oil is cold and thick, it eventually closes. But if it didn't, there would still be some oil pressure.
Most likely low oil level or debris/sludge in the oil pan. The engine
starts, debris is drawn toward the strainer, eventually plugging the strainer. When the engine
stops, oil in the engine, draining back to the pan pushes debris away from the strainer. It also could be the oil filter
if the filter type doesn't have a relief valve, for the same reason as the pan strainer.