This effect isn't going to be very significant. A constant exchange of air will have a significant effect, but a one-off exchange of 10% of the air will be negligible. The reason for this is that the air has a very small thermal mass compared to the building structure.
I've got mote sensors in every room in my house, including the hall. When my kids leave the front door open in winter for a few minutes, as kids tend to do, I can see the temperature in the hall dive on the graphs. But close the door again, and the temperature is right back up very close to where it started in just a few minutes - even when the heating is off.
The heat loss due to thermal expansion and contraction of air will be quite minimal.
Thermal density (specific heat) of air is going to be ~1000x less than that of solid objects.
The specific heat of gypsum (the primary constituent of drywall) is 1.09 kJ/kg.K
For dry air it's 1.0 kJ/kg.K
Air's density is 1.225 kg/m3.
A 6m x 9m x 2.3m (20' x 30' x 7.5') room has a volume of about 130m^3, so a 10% exchange would be 13m^3, or 16kg.
That's about 15 kJ of heat energy per degree C, or roughly 0.0004 liter (0.0001 gallon) of heating oil equivalent.
The drywall would be (in feet) 20x7.5x2 + 30x7.5x2 + 20x30 ft^2 (I'll assume the floor is some perfect insulator for now, and that the room has no doorways), and 1/2 inch thick, or 1.6 m^3. That's about 3600 kg of gypsum, which has a heat capacity of about 4000 kJ per degree C, or about 0.1 liter (0.027 gallons) of heating oil equivalent.
cheap apartments are also poorly air-sealed, which means they're already losing a great deal of their heat through air leaks. An extra one-time 10% turnover won't make a puddle of difference.
In winter in particular, the rate of through-roof air leaks is also greater when temperatures are warmer due to the stack effect. The author is completely wrong on this front - it's far better to let your house cool down and then re-heat.
The only time this isn't true is if you have a two-mechanism heating system such as a heat pump with resistance heat backup, where a large temperature swing invokes the more expensive resistance heater. The author doesn't.
I've got mote sensors in every room in my house, including the hall. When my kids leave the front door open in winter for a few minutes, as kids tend to do, I can see the temperature in the hall dive on the graphs. But close the door again, and the temperature is right back up very close to where it started in just a few minutes - even when the heating is off.