Tyre pressure and temperature in road cars

[NPP]

We all know about the importance of correct tyre pressure and of warming up the tyres in motor racing. I am wondering about the importance of these effects in normal road use. Of course, I don't want to sugggest that you warm up your tyres Alonso-style after leaving your driveway - rather, I am interested in the following real-life scenario:

The cheapest petrol station in my area is about 20 kms of motorway into my commute. That means that if I have a chance to check tyre pressures, it will be after travelling some time at about 150 km/h, mostly in a straigt line, with no hard cornering involved. How will this affect the temperature of the air in the tyres, and tyre pressure? Will the pressure be significantly higher than measured if cold? How would pressure and temperature be affected if it were 20 kms of twisty B-road, driven quickly, but within the limits imposed by law and by safety considerations? Will I have to put in an extra stop on cold tyres when I want to check tyre pressures?

Thanks for your thoughts!

NPP