Recoil impulse is simply the mass of everything exiting the barrel times the velocity it leaves (ie change in momentum). For the projectile(s) that is easy, for the propellant gases it is more difficult and usually estimated or directly measured in a lab.
Starting from Newtwon's first law we know that whatever momentum the projectiles and gases had leaving in the muzzle the firearm, the firearm must recoil with the same amount of momentum. Free recoil energy is calculated by taking this change in momentum, working from the assumption that the firearm is floating in zero-G, how much kinetic energy would the firearm have moving with that amount of momentum.
ie. A typical trap load might be 1-1/8 oz of shot pushed by 19gr of propellant to 1200 fps has a recoil impulse of approximately 2.82 slug-ft/sec (this is using the SAAMI's estimation that the average velocity of the propellant gases from a long barreled shotgun in this example is ~1.25 the muzzle velocity of the projectiles)
So if we fired this shell from a shotgun that weighed 7 lbs we would simply divide the recoil impulse by the mass of the shotgun (~.2176 slugs) to get the free recoil velocity of the shotgun ~12.97 fps. Once we have the free recoil velocity it is easy to calculate the free recoil energy using the kinetic energy equation E = 0.5*m*v^2
So we get a free recoil energy of 18.3 ft-lbs of free recoil.
So you can see if we fire both barrel we would double the momentum change since we would push twice the mass out of the muzzle end at the same velocity as shooting one barrel. But if we put twice the momentum change into our free recoil equations we see that we are going to double the free recoil velocity which when put in the kinetic energy equation result in 4X the free recoil energy.
Free recoil energy is a good number for comparing recoil as it takes into account both the cartridge being fired and the weight of the firearm firing it. The military uses it to limit how much solders can shoot various weapons in training.
