API RP934-A (based on ASME Sec. VIII, Div.2 design), para.5.5.1.1 is for tensile and impact test, not for step cooling test.
The step cooling test is performed on the weld metal which is based on heat of base metal, each lot of electrodes, heat of filler wire, and combination of lot of flux and heat of wire. The applicable components (i.e., thickness, shell, head, nozzle, pipe, fitting, flange, etc.) in API RP934-A are not clear.
In general, the detail requirements are specified in the company standards or P.O.

Here are some examples for the applicable components.

Company A : design code: ASME Sec. VIII, Div.1 through 3, design temperature ≥700°F and thickness >2 inches (Basic Concept)
Company B : design code: ASME Sec. VIII, Div.1 through 3, shell, heads, inlet nozzle
Company C : design code; ASME Sec. VIII, Div.2, thickness >2 inches