If we are just talking about the effects of casehardening or through hardening on a given double gun frame, no matter how high the hardness of the frame is raised, it won't change the amount of deflection of the frame when fired at all.

The stiffness of a gun frame will not change from hardness nor changing the steel from mild steel to chromoly, hard or soft.

The stiffness of steel is pretty much the same thru all the alloys and unchanged by hardness.