We had a massive power surge where the power blinked on and off rapidly more than 10x. It had a few times where it was off for about 20 seconds and then back on for about 10 seconds then off. Then it was off for a few minutes and came back on for a few minutes and then went back out for 6+ hours. Ever since then the AC hasn't been cooling to 70. The outside unit has been shutting off entirely. It didn't come back on after the power outage and I had to turn it to off at the thermostat and then flip the breakers (There are two marked AC). I did that, waited a few minutes, flipped them back on, waited several minutes and then turned it to Cool. 1st time compressor kicked on right away. 2nd time it took about 10min before it kicked on. Last time it seems to have kicked on right away. But it turns off after awhile and starts getting hot again. Not sure what the deal is. Guessing something got loose or fried a bit.
Sometimes when a unti won't turn back on right away it's because either the outdoor unit or the thermostat has a purposeful delay-on-make timer. This is to prevent damage to the compressor by trying to start against high pressure in the discharge line. More on that is below.
Other times, delay in starting is due to high head pressure still in the compressor discharge line. The compressor has a hard time overcoming that pressure. It takes a few minutes for the suction and discharge pressures to equalize. (The reason some units and thermostats have the delay mentioned above)
And, you could have a weak starting capacitor for the compressor. If you don't have a capacitance meter, a local motor repair shop will test it for you if you bring it to them. If you have two, it will be the round one.
CAPACITOR CAUTION!!! After you lock off power to the unit, wait a very long time (many hours) before disconnecting the capacitor. They store electrical energy for a living. Lots of energy!
The stored energy takes time to discharge. Some capacitors have bleed resistors across the terminals that will speed the discharge- IF THE RESISTOR STILL WORKS.
Or, some people short across the terminals with a screwdriver. If you're inclined to try this, wear full goggle protective eye wear, Sometimes capacitors explode, throwing shrapnel. I never do this and
don't recommend it. I have a bleed resistor with clips in the tool kit instead- along with properly insulated tools and insulated gloves.
Power Failures-
On-Off power is not good for the unit. Perhaps look into a whole home surge protector.
It won't help with dips in voltage, but will help catch the surge voltage when power recovers.
Paul