All wood structures sag eventually, even if they carry little load (such as your pergola). The only way to prevent sag is to use precambered beams that are specifically designed so that the eventual sag is approximately the same as the the camber, so after 10 years or so the beams are flat. This is expensive and totally unnecessary structurall, it is strictly an aesthetics thing, or in the case of bridges you want a flat deck, so the beams are precambered so they lay flat when the deck is attached.
You can compute the sag on your beams using standard beam theory and the dead weight of the beams, but they won't sag much, and since you are not putting any load on them (no roof, correct?) it is an aesthetics issue.