It's a balancing problem, fairly typical for a one-zone heating system spanning an entire house.
Mind you, I'm a bit concerned with a setup that kicks on the A/C in heating season - being from a primarily heating climate, we typically have a "winter/summer" or "heating/off/cooling" mode switch either on the system/boiler/furnace or on the thermostat to prevent that sort of foolishness. If it's REALLY so hot in winter or cold in in summer that we want to change the mode, WE choose to change the mode, rather than leaving it up to a machine to decide.
Presumably you have forced air heat, since the same system is cooling. One approach to improve balance is to run the fan more to distribute air around the house and balance temperatures even when heat is not being delivered. A more basic step is adjusting the airflow to different parts of the system for heat delivery that more closely keeps things even - but if the same system is cooling this may be difficult to get balanced in a manner that works well for both, since heat rises and cold air sinks, left to itself.
The system may be oversized (so it quickly heats or cools the location of the thermostat, and then shuts off the fan, rather than running a large percentage of the time when it's cold out), but all systems are prone to being somewhat oversized much of the year in order to be large enough to heat on the coldest days and cool on the warmest days.
A good HVAC professional may well be worth talking to in order to tune your system as best it can be tuned. Moving to continuous circulation (perhaps at a lower fan speed) seems like it might be needed in this house to reduce stratification, at a guess. The only major downside if your system is not too noisy will be the electric bill for running the fan.
As a quick stopgap, examine your thermostat to see if it has a "fan" switch, typically with two positions - Auto (blows only to heat/cool) and "on" or "Continuous" - if the switch exists and is wired correctly, that should put the fan in continuous circulation mode - but you may want to alter the system to make that quieter and/or more efficient with a lower speed or even an entirely different fan/blower. And it may still need to be balanced to work better.
Variation of the temperature from room to room that is different from the thermostat is a basic fact of operational behavior. you will have to get used to some variation!!
It is possible to even out the differences by changing how much of the heated air gets to each room. This is usually done by using vent controls on each room to permit more or less air into each space. Sometimes there are also vent pipe dampers or diverters that can be adjusted as well. A balancing exercise is usually an iterative process that takes time and patience.
A well balanced system will only stay that way if the same conditions apply as when the balancing was done. Some of these conditions would include outside air temperature, state of window coverings, whether doors are open, closed or partly ajar and if windows remain in the same state (closed or open).
Note that a system balanced for heating can take an entirely different set of adjustments to optimize for cooling during warmer parts of the year.
So as you can see you could try making small gradual changes in your setup. Try closing off more of the the vents that feed into the area where the thermostat is located and open those in the colder room a bit if not already fully open. Remember - iterative process.
Best Answer
Warm air rises, not vise-versa. If you are running the A/C on the first floor and leave the upstairs A/C off, then the downstairs unit should not be working all that much harder than it would if the upstairs unit was cooling the upstairs. So no, cold air is not drawn upstairs by thermal convection. That does not really happen.
Upstairs unit off and downstairs unit at 78 should reduce your energy bill versus running both the upstairs and downstairs units.
Edit:
There is an exception to what I said above. Because the upstairs unit must handle the extra heat load that comes from the sun shining on the roof, it is often the case that for equal square footage, the upstairs unit is larger capacity than the downstairs unit. Also, the cold air from the upstairs unit will flow down to the first floor via thermal convection (cold air sinks). If your downstairs unit is a bit undersized and your upstairs unit was meant to pick up the slack, then your downstairs unit will not be able to keep the downstairs cool all by itself, but this fact will not make your utility bill any higher than if the upstairs unit was running (unless your downstairs unit is also using large amounts of power from a malfunction and needs to be repaired).