Does a air conditioner shut off after it reaches the desired temp?

Most newer models do. In fact, most models that have a "set it to XX degrees" function do.

I recently installed a new 5 ton ceiling mounted ductless DX system in an electrical room It appears that the unit is over sized because the rest of the equipment will be installed within 2 years. My question is can I install a thermostat which has a higher differential temperature (may be 6) if there is any, or how can I solve this problem until the full load for the room is installed. I would really appreciate your answer.