I would suggest providing Temptale devices to the site (not very expensive), and these continuously monitor and can be downloaded monthly and printed out for you to view, or sent to you by PDF on email.

It is the sites responsibility to monitor the temperature and maintain quality of the IMP, have SOPs for quality deviations, temp monitoring and recall.

Usually it is the Sponsor who authorises the site to release IMPs after a deviation, after checking with the QA dept. (QP) that the stability is acceptable, particularly important is tracking multiple exposures to out of spec temperature as some medicines may not be reused after several exposures to incorrect storage conditions.
This can be risk asessed if you are using licensed products which are stored at room temperature for example, from experience Sponsors will (in this case) write clear instructions in a Pharmacy manual what to do in case of deviation of temperature, and under which circumstances the product may be released again for use, the robustness and documentation of these events is tested on inspection.

My opinion is based on experience from inspections and setting up the Pharmacy support of many many protocols