Probably should be in sample prep section but I will answer what I can.
A lot of how sample temperatures are done varies by the certifying agent as much as it does any methods.
For the current UCMR3 analysis the program states that samples are to be kept <6C (this come from the requirement of 4C +/- 2c) when in the laboratory and if arriving less than 24 hours from time of collection there must be solid ice in the cooler and the sample temperature must be 10C or less.
It is impossible to take a sample and bring it to a laboratory in one hour and the temperature of the sample itself to be at 4C, therefore most use a cooler thermometer to verify that the cooler itself is at 4C(usually with the +/-2c requirement, though some auditors hold to strict <4c but not freezing). The samples should be placed in a refrigerator that is kept between 0C and 4C which should be logged at least daily to ensure temps are held within spec. Most all the methods state to allow samples to warm to room temperature before analysis, except the new 524.3 which states the autosampler must be chilled and hold samples at <10C while on the autosampler awaiting analysis.
We did once have an auditor tell us that any sample received that was above 4C had to be rejected, but we were collecting samples a mile away that were coming from a holding pond. In the middle of summer you are talking about a water sample that is 30c! That sample will not be cooled down to 4C by the time it reaches the laboratory no matter how much ice you use in the cooler. The only way to stop getting violation notices from that auditor was to have that client place the samples in a refrigerator overnight and have us pick them up the next day. Makes absolutely no logical sense to do so, but it satisfied the lawyerish check list the auditor had