Will a water pump leak when the engine is cold?

I noticed my coolant reservoir was empty I filled it up and the next day empty again. I filled it up first thing in the morning before starting the engine that has been sitting over night. When I filled the reservoir I noticed a leak coming from under where the water pump is located. I couldn’t tell if it was... show more I noticed my coolant reservoir was empty I filled it up and the next day empty again. I filled it up first thing in the morning before starting the engine that has been sitting over night. When I filled the reservoir I noticed a leak coming from under where the water pump is located. I couldn’t tell if it was coming directly from there or not, could be a hose as well. Can a water pump leak with a ice cold engine? And would it drain the reservoir?
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