How quickly should oil pressure come up?

Earlier this year I switched from 5W-40 Valvoline synthetic "Euro Spec" to 10W-40 Penn Grade 1 for my Duster. It picked up 5-10 psi hot oil pressure at idle and above. According to the "ratings" it shouldn't have changed.
Au contrair. Based on the grade change, one would expect a possible if not likely change in viscosity at operating temperature.
Because in general a 5W-40 and 10W-40 will have viscosity curves that only overlap when the oil is around 212 F.
Then in addition, as Phreakish posted, the grades have an allowable range. Most manufacturers have spec sheets on their websites, and oil nerds will freqently post VOA and UOA viscosities.
The last piece of the puzzle is operating temperture. We can get a ballpark temperature based on coolant temperature - until we start looking at continuous high rpm high load. Oil temperature in the pan is a better indicator of oil temperatures in the engine.

Here's viscosity at termperatures from 32F to 212 F of four "40 weight" engine oils.
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Focusing on the temperature range one would expect for normal passenger car use.
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