ok I did a frame by frame count of the phasers from Star Trek 2 and Discovery. The rate of fire between them is pretty close. Though I can tell the beams travel at a faster speed in Discovery compared to those of Star Trek 2.
USS Reliant
4 shots / 1.2 seconds = 3.3 shots per second
4 shots / 1.1 seconds = 3.6 shots per second
USS Enterprise
5 shots / 0.7 seconds = 7.1 shots per second
12 shots in 1.66 seconds = 7.2 shots per second
5 shots / 0.7 seconds = 7.1 shots per second
(from 2 phaser banks)
Battle of the Binary Stars Ships from Star Trek Discovery
7 shots in 0.866 seconds = 8.08 shots per second
9 shots in 1.666 seconds = 5.4 shots per second
11 shots in 1.63 seconds = 6.7 shots per second
7 shots in 1.13 seconds = 6.19 shots per second
6 shots in 0.7 seconds = 8.6 shots per second
The rate varies