That sounds like a bad O2 sensor. Can you get any codes off of it? May be an O2 heater circuit or a worn out sensor? When it heats up and wont start does it still crank and just not fire? Other option is a bad CKP (crank position sensor), bad air gap, bad sensor or bad wire/connector has to be cold to make contact but heats up, resistance increases and bam. dead until it cools down again.