Google N-Gram shows that both "tempered distribution" and "temperate distribution" are used in English, but the first version significantly prevails, and usage of the second term declines.

Schwartz himself seems to have used this term for the first time in his paper of 1951, Analyse et synthèse harmoniques dans les espaces de distributions, Canad. J. Math. 3 (1951), 503–512, doi:10.4153/CJM-1951-051-5, and the French term was "distributions tempérées".

Google translates "tempérées" as "temperate".

I am not a native English speaker, but I understand "temperate" as a synonym of "moderate", which makes sense to me as a name of those distributions. For example, we say "temperate climate", not "tempered climate". While "tempered" seems to be related to metallurgy, namely to a process making steel hard. What does tempering of steel have to do with distributions?

Why does the name "tempered" win in English?