Accepted answer
Fourteen days at room temperature is about 47 refrigerated days of equivalent exposure. Room temperature is not a number, so take the pharmacopoeial 20–25 °C and its 22.5 °C midpoint: 17.5 kelvin above the 5 °C middle of a 2–8 °C refrigerator. The ten-degree rule of thumb — degradation rate roughly doubling per 10 K — puts that at about 3.4 times the refrigerated rate. It is an order-of-magnitude statement about a rate, not a shelf life, and the top of the 20–25 °C band runs about 1.4 times faster than the bottom of it. At 47 equivalent days a test is more likely to reassure you than to tell you something, which is a legitimate reason to run it and a poor reason to expect a finding. Either way, record the fourteen days and the temperature now, while you still know them; an excursion you did not write down is an excursion you cannot interpret later.
The honest answer is that the cold pack is doing much less than people think and that shipping lyophilised is what actually protects the material.
A single phase-change pack in a thin-walled box holds below ten degrees for roughly one to two days at twenty-five degrees ambient, and under a day at thirty-five. On a nine-to-fourteen-day lane the material is at ambient for most of the journey whatever was packed with it.
Stated carefully, a single-use temperature logger costs a few pounds, records the whole journey and converts an argument into a record. If the history matters, this is the answer.
Solid-state stability of lyophilised peptides at ambient temperature is documented in formulation studies with residual moisture as the dominant variable.
The caveat is that a warm transit is survivable for a solid and materially different for a solution.
Buy a logger if you actually want to know. Everything else is inference.