The single most important distinction is between what purity measures — the fraction of detected material that is your target — and what you actually want to know — how much of the material in the vial is your target.
Temperature affects the dynamics of molecular conformation, and if a peptide has proline residues that interconvert on the chromatographic timescale, the peak will split or shoulder at low temperature and collapse at high temperature.
Retention time is sequence-specific and method-specific, so comparing your result to a supplier value using a different method is meaningless without method documentation.
The Arrhenius relationship for peptide degradation is the basis of accelerated stability testing and also governs how quickly methods drift with temperature.
One qualification: achieving purity above roughly 98 per cent on a 30-residue peptide is fighting the chemistry of synthesis, not the quality of the purification.
Compare purity within a single laboratory on the same method, never across laboratories.