Will 2x10 13'6" span support outdoor bbq?
Yes, two 2x10's spanning 13'-6" will support 2,200 lbs.  Grade of lumber can be No. 2 &amp; better of typical framing lumber, like Hem.-Fir...it need not be Doug. Fir-Larch. In fact, each 2x10 will support 1,200 lbs.

To be clear, spans are measured as "clear spans". That is to say, from face of support to face of support. Also, if your BBQ is located in the middle of the span so that people can stand around the BBQ, then there will be an additional "Live Load" acting on the joists. This assumes the BBQ extends from support to support. 

By the way, the code doesn't require blocking until you use 2x12's. However, for this "concentrated" load, I'd use blocking. (Yes, I know, technically, it's not a concentrated load, because it extends from support to support. However, it could be considered "concentrated" because it's resting on only 2 joist, not over multi-joists, like in a large space (room).)

If the joists rest on a beam, then secure the joists to the beam with a clip. If they "hang" on a beam, make sure the hangers are rated for the load, because you're approaching the joists maximum load. The joists need a minimum of 1 1/2" bearing at each end. 

You didn't ask, but don't forget to install seismic deck straps (or rods) at about 20' oc. This is new to the 2014 code for all decks. 

Re: your question about doubling joists...will that double the allowable supporting load...yes, that is how it works. 

By the way, you reference a 10' space. Shouldn't that be 13'-6" space (span), or are you referring to the distance in the other direction. However, to me, that doesn't seem relevant because it doesn't contribute load to the 2x10 joists supporting the BBQ, right?