Does anyone know if studies show whether or not a 5 minute cooldown after a workout increase gains or decreases muscle soreness?

I don’t believe muscle soreness or DOMS has anything to do with a cool down. A cool down is intent on bringing the body back to its pre-exercise state gradually rather than abruptly.
As we know with orthostatis hypotension blood is distributed to certain regions of the body for specific reasons and a sudden change can cause light headedness because the blood has pooled and is not evenly distributed throughout the body.
I suppose it also allows contracted muscles to relax back to their normal state.