Even when the engine is cold the coolant has to be allowed to flow.
The pump for the coolant system runs directly from the engine, so is constantly turning when the engine is running.
If you took your thermostat out you'd see a small hole in the metal edge around the assembly of the thermo which allows coolant to flow.
Imagine you revving your engine with it cold forcing all that coolant onto a dead end, high pressure leak and potentially bang me thinks
So short answer, yes it should leak.