Temps, in both meanings, comes directly from two Latin words tempus, belonging to two separate declensions, which meant:

1. time (le temps qui passe); third declension (tempus, temporis)
2. weather (le temps qu’il fait); second declension (tempus, tempi)
3. the temple of the head (la tempe); third declension

So, the double meaning of time and weather was already present in Latin, though less marked. Going further from there would require a Latin etymological dictionary, which I don’t have available.

Il y a une très belle explication dans L’éclair immobile dans la plaine, philosophie et poétique du temps chez Lucrèce par Sabine Luciani. Cette explication est basée sur un article de E. Benveniste qui s’intitule Latin Tempus (ça ne s’invente pas ;)).

Ce lien google book permet d’aller voir la page 4 de ce livre.

In a literary circle that I attended after I asked the question, someone brought up the point the similar instruments, like sundials, were used by the “ancients” to measure both time and temperature. Also, there are similar words for the two concepts in non-European languages such as Japanese.

The closest relationship between the two that I could gather from the links that were provided was that time was related to the passage of seasons, and by implication, the weather.

From the above, it appears that “ancient man” regarded time and temperature as two sides of the same “coin,” even though this idea might seem ludicrous today. Hence, the common Latin etymology between the two, tempus, although the different declensions suggested that even they perceived a difference; e.g. that time led TO temperature changes. All this apparently occurred before the French language became popular.

The connection seems “counterintuitive,” but at least plausible given the above context. I would not have accepted the validity of a link to the Latin tempus without an explanation of this sort.

It should be noticed that in the Bulgarian language there exists -just as in French – one word with the same two meanings: време (pron. [vrème]). It means “time” as well as “weather”. So the explanation that the origin of he double meaning of the French word temps can be found in the Latin language, becomes more unsure.