Carbon dioxide pump is a disclosed pump used for pumping dense phase gases, specifically for injecting carbon dioxide into an oil or gas reservoir that includes a pump, a motor, and a casing in which the pump and motor reside. There is also disclosed pump used for injecting dense phase greenhouse gases into a reservoir or underwater. The pump may include a downhole electric submersible pump. It has a built in Peltier chiller and heat exchanger to prechill the fluid entering the pump and chill the pump head itself to a carefully controlled temperature. This pump employs dynamic compressibility compensation.