Well, you missed the fact that the plan is already to liquefy the air. So the air will already be in liquid form. Also I don't think they are going to cool it

Well, you missed the fact that the plan is already to liquefy the air. So the air will already be in liquid form. Also I don't think they are going to cool it but rather compress it. Cooling would require constant expenditure of energy to keep it cool whereas you can compress it and leave it in the vessel at room temperature. It takes 33 atmospheres to liquefy nitrogen and 50 for oxygen.