I have this on mower and it has been doing this for the last month.  I have done several things like drain the carb bowl but no luck.  Let it sit for 1+ hour and it starts right up and runs for 20+ minutes then dies.  Seems like carburetor but any other ideas? Strange that it would take that long for temperature to affect something.  Should come to operating temp much quicker than that.  
 

One of two things:
 
Either the ignition coil is failing, which is not un-usual..
 
When it dies, have a known good spark plug handy.  Plug it into the 
spark plug wire, ground to head and look for spark..  
 
Also, do you have a low oil shut off switch on that engine?  It'd have a thin, probably black,
single wire going to it.  If so, those are known to fail (especially on generators). They kill most 
of the spark...    
IF you do have it, just un-plug it to by-pass it.  It won't hurt anything but won't be a "safety" should you let your oil level go low...
 
Both of those cases either kill most or all the spark.   
 
Posting a full engine model # of the engine would help check the parts list to see if 
your engine is equipped with the switch...
 
.Please post back what you find..
 
.

thanks I'll look into the coil.  Was thinking that if it was failing do to heat it would have failed sooner but maybe not.  OI'll post the results.  Don't think this engine has a low oil cut off but I'll check that also.  
Joe