Yes, they will sag; technically any beam will sag, even under its own weight and on a very short span.
The qustion should be: how much sag are you prepared to accept, ie will it be noticeable?
However, I was just hair-splitting. Back on Planet Earth,I don't think you will have any practical problem with sagging (more correctly known in the trade as 'deflection').
(Where I am, our code allows 2x6 for floor joists at 16" centres over just short of a 10ft span, with low (ie un-noticeable) deflection, so your pergola beams will be just fine).
There might be a slight tendency to twist, so are you putting some blocking pieces between them?
I was planning to run 24" centers both parallel and perpendicular to the house, sort of a grid design. I planned on using a double 2x6 around the perimeter but I've been reading that I'd be better off using a wider board instead of doubling. I'm not sure which would be sturdier/ look better.