Accepted answer
The part that matters: freeze-thaw damage happens at the moving ice front, not at the storage temperature. Once the sample is frozen solid and cold, very little is happening. The damage is done during freezing and thawing, which is why the number of cycles matters and the duration of the hold mostly does not.
For the solid state, residual moisture is the dominant variable. A cake at two per cent water is considerably more stable than the same cake at six per cent, because water is both a reactant in hydrolysis and a plasticiser that lowers the glass transition temperature. This is why a desiccant in the outer packaging is not theatre, and why opening a cold vial in a humid room is a genuine error — you condense water onto the cake.
The temperature dependence is roughly Arrhenius over the range that matters, which in practice means every ten degrees of increase roughly doubles to triples the rate. Ten days at thirty degrees is therefore comparable to something on the order of a month or two at four degrees — bad, but not the catastrophe it feels like when you open a warm parcel.
General guidance on lyophilised peptide storage from the major synthesis houses converges on minus twenty degrees for long-term storage of solids and refrigerated storage for solutions in use, with the explicit note that repeated freeze-thaw of solutions should be avoided. It is consistent advice precisely because it follows from the chemistry rather than from a study.
One qualification: none of this addresses sterility. A vial can be chemically pristine and microbiologically compromised, and a chromatogram will not tell you which.
If the material arrived warm and it was lyophilised, test it and proceed on the result. If it arrived warm and it was in solution, the result is more likely to be interesting than reassuring.