Your A/C unit has a high pressure switch and a low pressure switch. If the freon pressure gets too high or low these will turn the unit off so it won’t damage its self. After the unit sits off for a while, the pressures equalize and it will try again only to kick off because of the switch again. You’ll need to have an HVAC tech look at it and hook up some pressure gauges to know for sure, but thats likely your problem.

You may indeed have a pressure switch causing this problem and in some cases a bad thermostat. I will add though that the vast majority of units made in the last twenty years do not have high and low pressure switches. Those safety components are only going to be found on the higher end of whatever product line you may have.

If the thermostat fan switch is set to auto and you do not hear the furnace fan short cycling on and off with the condenser outside than it is less likely a thermostat issue.

If you just installed this stat and are having problem with the set point. Some stats have a variable temp range. So for example lets say yours is 3 degrees. So if you desired set point is set at 78 you have a +/- swing of 3 degrees in temp. With that being said try setting it at a lower set point of say 75 and see if the problem persists. Does the blower motor come on in the furnace? If yes we can rule that out. Next would be if its a gas furnace the pressure switch maybe bad and that would cause the start and stop. But in that case the system would go into lock out mode.
If the thermostat has batteries in in and they are weak this could also be the problem.
Personally I think its the thermostat that maybe the problem. Like I said lower the set point of the cooling and try it.