Hi Luvyapop,
Refrigerators, by themselves, don't really use very much power as compared to what most generators put out. Things that normally test them are things like ovens, air conditioners, electric heaters, multiple electric burners, hair dryers and microwaves, in many cases combined together or with other things, etc.
What type of generator do you have? The make and model, if you know it, or just diesel vs gasoline.
Generators have a lot of safety shutdown. Not knowing the model, if it runs for 10 minutes, it is probably initially seeing proper oil pressure, and AC output voltage.
It could subsequently trip a safety shut down for a lot of reasons. It could be not cooling well and get over temperature. For example, maybe the thermostat is not opening. The oil could warm up and the oil pressure could drop. The load could be too much for too long tripping some other type of safety. Or the load could be too much pulling the voltage down. Just as some for examples.
The fact that it takes some time to cool down is interesting to me. It could be some type of mandatory time delay, e.g. from a time delay relay or IC equivalent. But, if it is many minutes, I'm thinking it could also, for example, need that time to self-correct the condition, e.g. cool down.
I'm also curious as to what it sounds like right before it shuts down. Does it just run nicely and then shut down like someone turned it off switch? (Could be a safety shutdown) Or does it seem to miss a beat or two before shutting down? (Could be air in the lines or something). Or does it surge and lag before shutting down? (could be dirty filters or a fuel line obstruction) Or does it seem to be labored? (Load?) Or does it seem to race? (Over RPM shutdown)?
I'd start with the basics. Is there enough oil? Is there enough coolant? Is water coming out the exhaust when it is running. Once running, if you have gauges, what is the oil pressure? Does it drop? If you don't have gauges, maybe get a cheap manual test gauge from harbor freight and have a look. Are the RPMs steady? If you don't have a tach on it, use a cheap phototach from Amazon (in fact, using one is generally a good idea, they tend to be more accurate). Put an IR thermometer on the block. Does it heat up? How high does it get? Does it reach a temp and level off or start to level off (like the thermostat is opening)? Or does it keep climbing? If it does keep climbing, how high does it get before shutting down? If you put a volt meter on the output, does it make full voltage initially? What about right before it drops? What color is the smoke initially? After it has run for a bit? As it approaches shutdown?
You get the idea. Basically, try to imagine why it might be shutting down and check that.