We should all be concerned with the sensitivity of cal constant to temperature change. Most measurements are done in the morning, so the temperature when the pre-cal is done is usually cooler than the temperature during the course measurement. This results in courses that are slightly longer than they need to be.
So for the most accurate measurements we would like to have the cal constant sensitivity to temperature changes to be as small as possible. There are many who suggest using narrow tires with high pressure will minimize cal constant sensitivity to temperature change. But the ideal gas law says the pressure change in your tire due to an increase in temperature will be the same regardless of the initial pressure and the initial volume of your tire. Using a narrow tire with high pressure isn't going to make your cal constant less sensitive to temperature change.
My experiments provide additional evidence that this is the case. The sensitivity of your cal constant to temp change is going to be pretty much the same no matter what tire width you use (assuming your wide tire still has a thin tread) and no matter what initial tire pressure you have. There really isn't a need to pump up your tire to 100psi every time you measure.
Tire "type" does matter. The thicker treaded tire in my experiments did have higher sensitivity to temp change than the thinner treaded tire.
I didn't test solid tires. And since the ideal gas law doesn't apply to them I have no idea what their cal constant sensitivity to temperature change would be.