The specific sentence in the question doesn't really make sense—it shouldn't have an indefinite article in the first place. (The definite article should also not be there.)
A better version would be:
✔ They assessed its durability in both hot weather and cold.
Although the indefinite article no longer applies, the second instance of weather can be omitted.
Here is a different sentence of the same type that uses the indefinite article:
✔ They ate a meal both hot and cold.
The syntax allows for an indefinite article. But note that it doesn't follow the exact construction of the sentence in the question. (The indefinite article comes before both rather than after it.) It's referring to a single meal that is both hot and cold rather than to one meal that is hot and another meal that is cold.
Finally, here is a construction that is identical to the one in the question:
✔ They ate both a hot meal and a cold meal.
→ They ate both a hot meal and
a cold meal.
→ They ate both a hot meal and cold.
I'm fairly sure that it's syntactically sound to omit the words that have been omitted. Since it's a parallel construction, the reader should be able to mentally replace them and understand the meaning.
However, these particular omissions seem a little awkward. At least stylistically, I suspect the omissions would not normally take place.