The wiring diagrams I have do not address the Hobbs meter, this is mostly ideas of where you might look. The Hobbs remote meter may be powered by the low oil pressure switch. If that switch stays closed when the generator stops, the meter will continue to run even though the generator is stopped. If there is a control board for the generator, the problem probably lies with the board, but you'll need the schematic to track it down. With the make and model number you may find the schematics online or someone here may have them and could send you an electronic copy. Sorry I can't be of more help.
Jim & Lin