Today's HalachaText said that clothing should be worn for at least 30 minutes. I asked them for a source and they said this number came from Rav Elyashiv Ztl.
שוב מצאתי I was looking for something else in Rabbi Simcha Bunim Cohen's Laws of the Three Weeks, and found that he quotes Rav YS Elyashiv there as saying a half an hour (page 85); in a footnote he cites Rav SZ Auerbach as saying that it needs to be worn "זמן ניכר עד שאינו נראה כמכובס" ("a significant amount of time, such that it's recognizable that the clothes are not laundered").
Footnote 140 there says that this is recorded in הליכות והנהגות (on עמוד ב? ...never saw the sefer, don't know how it works).
In that footnote, he quotes many other opinions about the length of time for wearing clothes to prepare for the nine days. שו"ת רב פעלים (chelek ד, end of סי' כ"ט) says the time period is 2 or 3 hours. קרא עלי מועד (page 26, footnote ז) quotes the above opinion of Rav Elyashiv at half an hour.
On the shorter side, דעת קדושים (YD 389:3) says that that wearing it for a minute suffices ("דאפילו לבישה לרגע מהני") [though others question this opinion strongly].
משפט צדק (responsum 46) says that wearing it "an hour" before the nine days only helps for the time in between Rosh Chodesh and שבוע שחל בו (because this is just a custom); but to wear clothes from the beginning of שבוע שחל בו wearing clothing just for an hour is not enough (because this is מדינא).
And no, DoubleAA, still no mention of climate/temperature....though I did not look up these sources myself, they are just quotes of quotes (of quotes?) from the footnote of Laws of the Three Weeks.....though they may help you find the information you want by looking into them....