GFI

GFIs are very sensitive. They also wear out. It's hard to tell why that happens to you, however, since we cannot examine first hand the barbecue or the GFI.

Or you could have a short in your BBQ but not a bad enough short to trip the breaker. If possible, try to run the BBQ using an extension on a different GFI circuit. You probably have them in the kitchen and bathrooms as well as the outside plugs. You could also try to run a similar sized load in on the GFI in question to see if any large load trips it.

Is the BBQ using the the plug for heating element or to spark the fuel?
Either way there would be a difference in the current between the hot and neutral which would potentially cause the GFCI to trip. You don't usually put heating loads or motors on gfci's for this reason.