I have a White Rodgers thermostat series 1F78-151 and It doesn't work when I put it on cool.

You need a replacement obviously. this one is cheapest I have found. take a look: White Rodgers Programmable Thermostat 1F78 151. I hope this was helpful. cheers

- Check the wires on the terminals make sure that they are not loose
- Reset the unit by press time, up and down buttons make sure the systems is in on position.
- Set the system to cool then and lower the setpoint below room temperature. There may be up to five minute delay before the thermostat clicks in cooling. If the thermostat click this means its not a thermostat you have a problem with your cooling equipment call your installer. If the thermostat did not click the unit should be replaced.

Check the compatibility of thermostat & how to wire thermostat .
I hope this was helpful :)
If not, you should replace your thermostat. White Rodgers 1F86U 42WF Thermostat is a good choice ;)

Why not buy the most popular thermostats like Nest or Ecobee?Rodgers is also a good choice, but their support is not so good.

The problem you meet may caused by lines miss connect, you'd better reconnect the lines. If you want buy a new one to replace the Rodgers, you can see reviews here, 6 Top Best Wifi Thermostats Best Wifi Thermostat Reviews