When boiling water at room temperature, what is the relationship energy has with the air pressure that is making the water boil cold instead of hot?
Think of it as the air pressure holding in water, preventing the water in the container from reaching a lower energy state. Without pressure, the fastest-moving water molecules fly off, leaving behind slower (i.e. colder) water.
As water boils, disregarding superheating, http://en.wikipedia.org/wiki/Superheating, the liquid stays at the boiling point and energy from the heat source goes into breaking the bonds (e.g. hydrogen bonds) holding the liquid together. If pressure is lowered, the heat in the water can vaporize more of the liquid, removing the fastest molecules, thereby cooling the remaining liquid to the new boiling point.
Removing air pressure is not adding energy to the system, though work is done against atmospheric pressure. If you have a piston between two cylinders, as it moves away from one side, pressure drops, but work must be done to compress the fluid on the other side.
Boiling occurs when the vapor pressure of a liquid reaches ambient pressure. In the example you asked about, water can be made to boil by removing the air in the chamber in which the vessel holding the water is kept. When the air pressure gets low enough, there is not enough pressure to keep bubbles of water created by the water's vapor pressure from expanding. We call this boiling. At atmospheric pressure the water still exerts vapor pressure but the air keeps the bubbles from expanding. The only way for water to reach the vapor phase here is for it to evaporate from the surface so we see no bubbles.