how to switch on a inverter when my fridge thermostat switches on
simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. Control circuit for inverter switching.

This does not have a time delay. If you really need that then get a mains time-delay relay and add it between the inverter and the fridge. I would have thought that a 1 s delay would be more than enough.



With the added constraint that the fridge must be unharmed so that it can be run on mains:


The additional 12 V T'STAT will need to be mounted inside the fridge.
Set T'STAT to the desired temperature and set the fridge thermostat to minimum. That way the fridge will be controlled by T'STAT rather than the fridge stat and it will ensure that you won't be running the inverter while the fridge's 'stat is open.
When running on mains set the fridge's 'stat to normal position.