What could cause the APU on a Boeing 777 to fail and also trip the GPU?
Dispatching an aircraft with an inoperative APU, regardless of whether it's a passenger aircraft or a freighter, is a normal procedure. While an APU may be started inflight preparatory to it's use after landing, it is not needed inflight, so it's not a safety issue. In my experience (retired in 1999), dispatch of freighters with inoperative APUs is more common than dispatching passenger aircraft that way due to the passenger comfort aspect. Keeping passengers cool while unloading/loading is a priority. Keeping a freighter's crew comfortable is not.

An APU failing while in use is not unusual. GPU failure is also not unusual. If the APU tripped (as opposed to just quit) and the GPU tripped as well, a natural suspect is an electrical problem in the airplane. What is needed, of course, is a competent mechanic to troubleshoot.

APUs can fail for a number of reasons. They're just small turbine engines. In fact, for 747-100 and -200 aircraft, the APUs were often the same as the engines used for SA-226 and -227 Metroliners (with the addition of propellers of course) that I flew prior to flying 747s. Common causes of APU failure include overheating, loss of oil pressure, excessive load, and just plain mechanical breakdown.

The decision as to whether to dispatch a large airplane without an APU would be an economic one primarily. If you're sending the airplane to an airport that is known to have GPU power, it wouldn't make financial sense to delay the flight because of an inoperative APU. If, however, you're dispatching to an airport that doesn't have such, that would be unwise unless such an airport does have a huffer and you want to risk a battery start (never a good idea in my opinion).