This will help prevent damage to your system and help keep you safe. When the nest is cooling the fan or whatever moves air through my vents will turn off every 2 3 minutes before reaching the set temperature but the ac unit itself sounds as if it is still.

Nest Protect Review 2015 Improved Safety Makes It Easier
Nest keeps turning on and off. Unless the air handler is turning itself off to protect itself i think my nest is defective. Turn the ring and select reset. Turn off power to your system at the fuse box or system switch. On the ecobee3 turn off smart homeaway in the settings. If you have your old thermostat hook it back up and try that. Nest keeps turning off and i have to pull it off the base and recconect.
To turn all motion sensing features off on the nest just turn off auto away and homeaway assist in the settings. Check that all connectors. Can you please turn off the wifi capability of the nest thermostat and see if we can get a a stable battery level. However after several minutes it started cycling again at around 2 3 second intervals. Dont have old and if youre feeling diy enough. You can disable power connect yr and g if you have it to each other and then energize it.
Pull off the nest thermostat display. To reset first find the mic button on the side of your nest mini and turn it off. Now press the very top of the nest. The led lights on the top of the mini should turn orange when this happens. Need to determine if its the ac unit or the nest. Last night i successfully got the fan to stay on for several minutes by having nest tell it to stay on.
Press the thermostat ring to bring up the quick view menu. Whatever you decide to do you at least now know what the likely cause is whenever your thermostat decides to take a long break even when youre home.