We just moved into a rental home. There is one HVAC system, but 2 thermostats. We noticed that when we are upstairs, it sounds like the unit is running constantly. We cant seem to get the temperature right in this house.
Is the house drafty? We had a similar issue with the first floor, the heat always turned on and off. I first replaced the old thermostats for new ones, but still no affect. What it came down to was our old windows created a lot of drafts and the door to the outside was right next to the thermostat, so every time we opened the door to go outside and cold air entered it turned back on.
I would check the locations of the thermostats, and potential influences to cause the temperature around them to be effected.
Also try adjusting your thermostat to reflect the overall temperatures of each floor. With our home upstairs is always 4 degrees hotter, and so if we set the 2nd floor to 65 we set the 1rst floor to 61.
Just because it has two thermostats doesn't mean it's dual zone. I've seen people add a second thermostat that works in tandem with the original but the AC was not converted into a dual zone. So either upstairs or downstairs can turn on the AC/Heater but it will cool/heat both floors even if both don't need it.
Dual zone requires an AC unit setup for it or an additional control board to handle the two thermostats plus automated dampers to control the flow to each zone.