Damage is listed as 4x20, for example.
Damage per second calculates in all of the relevant damage stats, including multiple barrels.
So a quad barreled gun that does 4x20 damage, and 200 damage/s, will deal 80 total damage each volley, while doing a total of 200 damage a second.
You don't need to adjust the damage/s tooltip unless you're factoring in overheating or energy usage, but then it gets complicated.