1st, try another gauge.
Next, increase the viscosity to like 10W40, get the temperatures to match and see what the pressure does.
If it goes up, the pump gear end play may be loose (over .003") or too much gasket thickness between the body and cover, but the relief valve is OK. If this oil is good for you (cranking speed is fine, pressure goes up instantly when started) I'd feel good leaving it.
If it doesn't go up, try even higher like 20W50 or straight 40 (no hard throttle until it's warm). Same deal - if it goes up, either the pump or bearings are loose. That oil isn't really a great choice for all-around.
No pressure increase means the relief valve is blowing off too low - bad spring, leaking relief piston. This means the pump has to come out.