switch pump on with transistor, mosfet, or relay
Transistor - Big voltage drop between C-E @ 5A (depends on type but it will be about 2V). Power dissipitation is voltage drop times current so about 24Watts. You must be careful with inductive load - induced high voltage must be supressed. Usual way is using anti-parallel fast diode over load.
Mosfet - Voltage drop can be really small, so it might be possible to use it without heatsink. Big Gate capacity needs strong driver to charge and discharge. If not, slow switching causes huge losses (again voltage drop times current). For inductive load it's same as for transistor.
Relay - It must be selected carefully as switch-off causes electric arc (= burning contacts) and because of DC current it's burning for much longer time than AC.