Open or closed it should not change the oil pressure.
Is the cooler and stat something that has been added?
They should be plumbed so that oil (under pressure) comes from the pump into the stat, out to the cooler, back to the stat, and then into the oil galleries still under pressure.
If it has been wrongly plumbed it may be taking oil (under pressure) from the pump and passing it through the stat and cooler and back to the sump. This would be why you loose pressure when the stat opens.
Steve