Looking on the internet deeply has found these results:

But the same time is the most popular phrase on the web.

Dhj? 130,600,000 results on the web	But the same time 133,300,000 results on the web More popular!
Some examples and use cases from the internet:	Some examples and use cases from the internet:
It doesn't look like this phrase is very popular! The search could not find any examples on the internet.	But at the same time I'm relieved. But at the same time, it was... Strangely reassuring. I see it as two steps: first and foremost that we support the Committee on the Environment; but the same time it would make sense to go a step further and support the amendments which have been presented to the parliamentary debate by the Green Group. But at the same time, they bring in these epic mythology story lines... But at the same time they both had troubled children... But at the same time, he wants to be anonymous. But at the same time, his money finances packages... that he himself will never touch. We must save as much electricity as possible... but the same time raise production.

Dhj?

130,600,000 results on the web

But the same time

133,300,000 results on the web

More popular!

It doesn't look like this phrase is very popular! The search could not find any examples on the internet.

But at the same time I'm relieved.
But at the same time, it was... Strangely reassuring.
I see it as two steps: first and foremost that we support the Committee on the Environment; but the same time it would make sense to go a step further and support the amendments which have been presented to the parliamentary debate by the Green Group.
But at the same time, they bring in these epic mythology story lines...
But at the same time they both had troubled children...
But at the same time, he wants to be anonymous.
But at the same time, his money finances packages... that he himself will never touch.
We must save as much electricity as possible... but the same time raise production.