2003 Hyundai Tiburon Starting (cold start)

This is actually pretty common on all car brands. There are two sensors that often fail by becoming heat-sensitive, then they'll work again once they cool down. They are cooled by natural air flow while you're driving, but when you stop a hot engine, they get hot from "hot soak", meaning the heat from the engine migrates up to the sensors rather than being blown away.

Start by having the diagnostic fault codes read and recorded. Many auto parts stores will do that for you for free. If there are no fault codes in memory, you'll need a scanner to view live data and see which sensor signal is missing during the no-start condition.