It depends on where and how you set the tyre pressure, and what the temperature difference is. Most pressure gauges measure difference to ambient, not absolute pressure.
If you set the pressure at 2 bar at 2500m, this mean the pressure in the tyre is 2 atmospheres above the ambient pressure at that height, which is around 11psi, so you'll have 22psi in your tyres. Set it at 2 bar at sea level, and you'll have 28psi. If your gauge is calibrated to sea level (which it almost certainly will be)and you use it at 2500m, you'll think you have put in 28 psi but actually you'll only have 22psi.
Come down to sea level and (excluding the effects of temperature) you will still have 22psi in your tyres and the gauge will show this, but of course ambient air pressure will now be 14 psi so if you want 2 bar in your tyres you'll need to pump them up to 28psi.
But since PV/T = Constant, if the temperature has gone up, then so will the pressure, since car tyres can be considered to be reasonably inelastic.
So if you've gone from say 0C to say +10C as you come down the mountain, your tyre pressures will have increased by approx 2psi, partially offsetting the decrease - at least, that's what I always understood, pressure increases around 1psi for every 5C increase in temp.
Sorry, I disagree (partially). I don't believe that pressure gauges work by giving multiples of the local ambient atmospheric pressure; they are after all a variation of a diaphragm versus a spring. So, imho, 2 bars at 2,500 metres is 2 times 14.5psi (not 2 x 11); then the absolute pressure in the tyre would be 29+11 = 40 psi. From here on I agree with you, although my conclusion is slightly different with the figures I used:
Come down to sea level, the absolute pressure will increase due to a temperature change of 16 deg (= 2.5km x 6.4degC/km altitude). Standard sea level temperature is 15C = 288 Kelvin. Therefore new tyre absolute pressure is 40x288/(288-16) = 42.4 psi. So now the gauge pressure will be 42.4-14.7 = 27.7 psi, a decrease of 1.3 psi.
Or putting it another way, your tyre pressures will have dropped by about 1.3 psi when you reach the top of your Alpine pass - hardly worth bothering about.
Sorry about the nit-picking!
