The relevant detail is that 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.
The fraction of your main peak that is actually your target versus isomers, fragments or related sequences is invisible without complementary identity testing.
The underlying point is that 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.
The ICH Q3A impurity thresholds and the relevant pharmacopoeial chapters all specify method validation requirements that almost no research-grade certificate claims to meet.
I would be careful about over-reading a single measurement — treat it as a data point, not as ground truth.
Compare purity within a single laboratory on the same method, never across laboratories.