The walls of your house are touching the outside air and will be at a temperature between your inside air (65) and the outside (hot or cold). Everything that has a temperature will emit some heat as radiation, mostly infrared, and how much is determined by the temperature of the object. You and your walls are always exchanging heat this way and when they are colder you feel less of this radiation reaching you making you colder. I did the math at one point but I don’t remember the exact answer anymore. It’s noticable.
There are done good answers here. My guess is humidity. Where I live (where winter means negative C temperatures for 4.5 months) our houses get very dry in the winter with the furnace going. Humid air holds heat energy in the water in the air. Dry air holds less heat energy. Evaporative cooling occurs more readily when air is dry. So the air temperature could be the same but you would feel evaporative cooling as moisture leaves your skin and evaporates into the dry air, taking heat energy from your body and putting it into the air. I could be way out to lunch but that is my theory based off what I know.
Relativity
If you were out in negative 20 and came into 65 degree room you’d appreciate it more
Same if you were out in 110 and came into 65 degree air conditioning
It’s not the 65 degrees that feels colder but what’s going on around it
If you had a 40 degree day after having below freezing for months you’d feel like it was tshirt weather
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