I flew the 737-200 for some years in the tropics and I saw that the altimeter, despite being fed by an ADC, had a consistent error, that I decided was due to the above ISA temperature. It is not dependent on the temperature on the ground, or at the airport, but the ambient temperature that the airplane is experiencing at the time.
Obviously there is no need to make any corrections for high ambient temperatures.
After some time I started to fly the B757 over the same routes and I saw that there was no longer a consistent altimeter error. Since I had already decided that the 737 error was due to temperature, this threw my theory off, so I checked the stats for the ADC and found that it did, in fact, correct for temperature. In other words the altimeter reading is for True Altitude. (but this was a long time agon in Pilot years and I do not have reference to those manuals now).
I have assumed the same for all subsequent airplanes I have flown and my observations tend to support this view, whether in expremely cold or hot conditions. Again, it is not dependent on the ground temperature, but the ambient. Using Ground temperature to calculate corrections for the approach is a crutch, as you would see if you examined it closely.
Never mind all that, is the correction important? And if it is, how is it applied? You do not want to go around the pattern adding 500 or 1000 feet to all the ATC assigned levels, or else YOU would be a danger to others who are not doing the same.
It is important that the correct altitude be used for the approach minima, and so a correction should be applied to that, but an examination would show that it is usually less than 40 feet even under the most extreme conditions for an ILS, and maybe double that for a non-precision approach.
How would you determine the error in practice? One way I have found to be effective is to check the altimeter at the OM or DME fix on a Glide Slope, and as I said, the later generation airplanes seem to be "spot on" at this point, while the older airplanes had an error equal to the temperature correction.
So, to follow my own advice, I went to the Techies. If you care to look at the Boeing Maintenance Manual B747-400 (one source), on Page 24 of Section 34-12-00 there is a statement (10.a) that says "Pressure altitude is derived from the static pressure input which is compensated for ambient temperature and corrected for static pressure source errors."
I do not rest my case, since there may well be other documents affecting this. in particular the way the data is transferred and customer options, and it may not be applicable to Airbus airplanes. The manuals for the ADC manufacturer details the way the pressure and temperature values are related, but it is in techie language and makes little sense to me.
Making the temperature correction, whether required or not, will not get you into trouble, so please do what YOU believe is right.