Accepted answer
Stated carefully, 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.
Gradient slope controls resolution, and gentler slopes resolve co-eluting impurities into separate peaks — so the better method reports the worse purity number.
The Arrhenius relationship for peptide degradation is the basis of accelerated stability testing and also governs how quickly methods drift with temperature.
Compare purity within a single laboratory on the same method, never across laboratories.