room temperature is above the refrigerated band, and how far above is the whole answer. 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. Lyophilised material tolerates this far better than solution does, which is why it is shipped that way, but "tolerates" is a statement about weeks and the vial is being asked about months. Log the excursion rather than guess at it. Nothing here is medical advice, and research-use-only material is not approved for human use.
Answer first: dry, cold, dark and still, with the lot code where you can still read it in a year.
Let a refrigerated or frozen vial equilibrate to room temperature before opening, so humid air does not condense water onto a cold cake.
Keep vials in their original carton or in an opaque box. Light exposure matters for tryptophan-containing sequences and for anything with a chromophore, and a closed box is free.
Two-to-eight-degree refrigeration and minus-twenty freezing are the standard storage conditions for lyophilised peptides in supplier documentation across the market.
The caveat is that storage conditions preserve what you have and cannot improve it.
Equilibrate before opening or you condense water into the cake.