stumbling on this one. Totally agree that "au même temps" is grammatically incorrect. I've NEVER heard it, ... except by little ones. ;-)
It may be a local accepted variant though (kind of "patois") in some areas.
And from your quotes and old french books, it looks like it used to be correct a couple centuries ago. A lot of phrases and words were used differently then though.
I agree also with previous answer, that it would work in a couple very precise cases, if "temps" is a noun as in "present tense/ past tense" ( = au même temps") or in music (as in pulse/beat I guess... although it would have to be really specific cases to make it work)
A la même période can be a correct use for cases like "we were in London at the same time" = on était à Londres en même temps / à la même période.
I have a native French teacher who always corrects me when I use "au même temps" in papers, but for some reason, I always want to say it! I avoid it, just in case, but this is an interesting discussion.
For those of you who argue about "au même temps", whether it is correct or not, I would simply say "au même moment" and avoid all the confusion from now on, so basically we have "au même moment", "en même temps", and "à la fois", though the latter is rarely used.
As a french native. "au même temps" means "au même moment" or "au même instant"
Here "temps" is the idea of a unique period of time. It is understandable when you want to describe what is happenibg during this same period of time, you might find it in poetry and in technical text but I confirm it is never used in everydaylife. In a physics class book yes but not everyday french.
"en même temps" means simultaneously, in the same rythmes.
radom example : "et si on chantait en même temps?" gives "what about singing together?"
hope it'd help
I'm afraid some people might misunderstand this...
a) It's a quote by Gide, more than a century ago, and Gide is known for his very old-fashioned style
b) Above all, temps is used here in the meaning époque, as I understand it means à peu près à la même époque
Anyway, Gide or not Gide, I would definitely put you off saying au même temps today in 2015 !
I guess it's quite clear from my post that the phrase doesn't see much use in modern everyday French. Besides that I also looked to rectify what had been said earlier, namely, that the phrase "au même temps" was ungrammatical.