Accepted answer
Mechanically, purity is a method-dependent figure, and that is not a limitation of the measurement, it is a property of what the measurement actually answers.
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.
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.
Published side-by-side method comparisons show that a two-point difference in purity on the same vial is easily explained by method choice alone.
Worth noting that method standardisation is poor in the research-grade space compared to pharmaceutical work, so identical-looking methods can produce different results.
The practical summary: ask for the chromatogram and the method, and ignore the headline number until you have both.