Your description is not very clear, but here is what I think you are doing. When your thermostat reaches its threshold it makes a contact closure or energises its internal relay. AT that time you need to provide a 2-second pulse to the cooling unit to start it. At the end of the two seconds, the start pulse ends and the cooling unit continues to run until it brings the temperature of the thermostat back down and turns off. You now are waiting for the thermostat to warm up and start the cycle again. So basically you need a motor start-up pulse.
If this is correct, you just need to control the relay from a one-shot. You will need a relay to provide the power to the cooling unit. You will need a one-shot such as the LM555 to provide the timing. You will need a power supply of 5-15 volts.
You will need a reverse diode across the relay coil. (1N4002). You will need a transistor from the output of the LM555 to use for turning on the relay. You will need a decoupling capacitor, an input noise resistor-capacitor, and a timing resistor-capacitor.
Search application notes for the LM555, MC1455, UA555, LMC555, and I think you will find a suitable application note.