You said it starts dripping when the pressure builds back up. Why is there pressure building if the temperature is stable and there is no fermentation activity?
I would think that in most normal circumstance your barrel and tap should not leak. If the pressure is not that great then you might have a faulty barrel/tap.
In any regard you could still install an airlock to relieve pressure.
As far as "cooling it", in general you want to store the cider in a stable, cool environment. So if it is in a warm area of the house, put it in the basement or cellar or even in the icebox.