Accepted answer
37 °C is above the refrigerated band, and how far above is the whole answer. 37 °C is 32 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 — makes that about 9.2 times the refrigerated rate, which is an order-of-magnitude statement and not a shelf life. 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.
Start with the physical state, because lyophilised and reconstituted material have completely different storage answers.
Store with the desiccant sachet if one was supplied, inside the secondary container rather than loose in a drawer. Residual moisture is the dominant variable in solid-state stability.
It helps to be literal here: let a refrigerated or frozen vial equilibrate to room temperature before opening, so humid air does not condense water onto a cold cake.
The 28-day in-use period for preservative-containing multi-dose preparations is printed on the diluent labelling and is a long-standing pharmacy convention.
Nothing here is medical advice, and research-use compounds are not approved for human use.
Keep the desiccant with the vial and the certificate with the lot code.