HVAC thermostat issue?

I moved to a new house (4 bedroom, almost 2400 sqf) few months ago. The house has a relatively new Bryant HVAC and as the weather is getting colder we decided to turn on the heater. The thermostat is programmable and we set the temperature on 67 in the early morning (6-8 am) for 2 hours and also 67 for 8-10 pm.... show more I moved to a new house (4 bedroom, almost 2400 sqf) few months ago. The house has a relatively new Bryant HVAC and as the weather is getting colder we decided to turn on the heater. The thermostat is programmable and we set the temperature on 67 in the early morning (6-8 am) for 2 hours and also 67 for 8-10 pm. For the rest of the time , it was set on 60.The HVAC somehow works most of the time with a very short breaks (off time) in between (may be only 5-6 minutes and sometimes 10 minutes), however the temperature stays at 67. I checked the temperature with another thermometer and the thermostat shows the correct temperature. The HVAC also starts to work 1-2 hours before the time that we set it to warm up to 67 (despite the temperature is above 60, probably at 63-64). Our last gas bill also was significantly higher than before.
Question:
Is there anything wrong with thermostat or the HVAC/system? I had two professionals to come in and check the system, one of them told me that everything is okay and the heater just works constantly as we lose the energy due to the big house and poor insulation. The second one told me that the thermostat has a problem and needs to be changed!

I appreciate your input.

Thank you for your attention and time.
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