I would gather up a bunch of blow dryers and electric heaters and add them up until I got a resistive load about equal to the rated output of the generator head being conscience of equaling distributing the load on both legs of the output and see if it can produce the rated output. If you have a Kill A Watt meter or something that to verify the loads that would be a plus.
Check the oil level. Even if it was recently serviced that does not necessarily mean the oil level is not low. If it is a slanted oil fill hole fill it to where it is almost running out. You can temporarily disconnect the low oil float switch to check for an intermittant problem. It is usually a yellow wire.
Visually check for any wires (probably black) or connectors that might be making contact with ground due to engine vibration.
Sometimes a bad On/Off switch can be intermittant. Most On/Off switches are open while running and connected to ground to turn off the engine. Disconnecting it temporarily (probably a black or brown wire) may help locate an intermittant switch.
Remove the fuel line and check to see if it is flowing easily (use a suitable container to catch the gas, of course). If not remove the fuel filter (if it has one) at the tank shut-off valve and clean it. Honda sometimes uses a filter screen inside the fuel line but probably not on the GX engine.
Check the air filter.
Check the governor to make sure it is moving freely.
Check and regap the spark plug.
Sometimes a bad ingition module will start and run just fine and then suddenly stop. Then it will start again.
If you do all this (what I call routine maintenance) and it stll has problems remove the float bowl from the carburator and check to see if the float is stuck. Sometimes there will be a little burr on the edges of the float pivot that can be gently sanded off. There should be a screw near the bottom of the float bowl for draining out the gas before taking if off. Turn off the fuel supply first, of course.
Are you doing anything such as connecting this generator to an inverter such as an OutBack or Xantrex SW?
Maybe the generator doesn't like your inverter AC. Does it work OK with other comparable loads? None of my microwave ovens work well with any of my generators. Neither do battery chargers.
Is this thing still under warranty?