Bluskis
Altimeters and ASI’s are both designed to react to pressure changes only.
i.e. if the static pressure is A and the Pitot pressure is B then the Altitude will be C and the IAS will be D etc.
Neither instruments ‘cares’ what the temperature is. If as in Huggy’s example a local temperature drop has resulted in an increase in pressure then the altimeter is designed to react to the pressure increase by indicating a lower altitude. That is basically the definition of pressure altitude.
One parameter where we do require temperature to be incorporated in our calculations is True Airspeed (TAS) . To calculate TAS density must be known and this can be calculated by modifying IAS (or CAS to be precise) with temperature. This is the reason that larger aircraft have a Total Air Temp (TAT) probe.
Having said all of that all of these instruments will have some internal temperature compensations that are purely for MECHANICAL REASONS e.g. a lever may be a bit longer when hotter, a capsule may not have the same expansion characteristics at one temperature than another.
It is easy to get the two types of temperature compensation muddled.
In the TAS example all TAS indicators (or more usually an Air Data Computer) will calculate TAS from CAS using the same calculations derived from the Standard Atmosphere model.
In the case of compensation for mechanical reasons individual instruments will differ in the amount of compensation required depending on construction methods material used etc.
Hope This Helps
Max