Actually, the evaporation effect will reduce the temperature of whatever is around it, in this case the air. I don't know whether the effect is dramatic enough to provide significant cooling for an entire apartment, but it's probably enough to make you feel it. Do a calculation using the heat capacity of water and referencing it against how often you have to refill the bucket.

Evaporation is what separates nice, cuddly good-weather clouds from thunderstorms.