I'm not that clued up on cruise but I would reckon from the symptoms that it must be a wire/switch/sender that gets hot - i.e. its on the engine. Its getting hot as the engine runs up to temp then it goes open circuit and whole lot stops - this is borne out by the fact that it runs longer on cold days.
So look very closely at anything related to it ON the engine - a circuit diagram will come in handy.....
Bri
