oil stats work in a progressive manner the same as the one in your engine water cooling system.
Constantly varying flow to the cooler, not on off.
I think any engine, regardless of ambient temp, when fitted with a cooler, needs to be brought up to operating temperature as quickly as possible and then kept on a constant to minimise engine wear.
Fitting a thermostaticaly controlled oil cooler is a must in my opinion on any Turbo or super charger converted engine especially, for the obvious reasons of additrional thermal loading and is beneficial in helping prolong turbo and charger bearing life, through helping maintain a constant oil flow rate from minimising temperature related viscosity changes and minimising clearence fluctuations due to component temp variation.
Cheers
Trev