I work as air traffic controller in Asia and we refer to both ICAO and FAA regulations. As winter is coming, there is a debate on altitude correction (or altimeter correction?) for cold temperatures. Some insist that we (ATC) don't have the responsibility to make altitude correction, and the others insist that we have, when ATC performs radar vectoring. (We know that the pilots have responsibility when IAP.)
In related to this, ICAO DOC 8168 states (Vol 1, Chap 4 Altimeter correction):
The pilot is responsible for these corrections, except when under radar vectoring. In that case, the radar controller issues clearances such that the prescribed obstacle clearance will exist at all times, taking the cold temperature correction into account.
So, I've tried to find references about this but failed.
Here're my questions.
1) Is there a document or manual for ATC about how to make altitude corrections when radar vectoring?
2) Why is there no altitude correction for hot temperatures that make for a steep angle of descent?
3) Should ATC-issued altitudes take into account temperature-induced altimeter error, or does should ATC issue the true altitude and leave the responsibility on the pilot for making the correction?