Looking on the internet deeply has found these results:

at the cost of is the most popular phrase on the web.

at cost of 16,710,000 results on the web	at the cost of 1295,000,000 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 it can't be at the cost of the other things we are trying to achieve here. Once students' rights come at the cost of others, it is my job to intervene. This option renders a better image at the cost of increased calculation. However, this must not come at the cost of social uncertainty. financed public investment will produce a short Instead, debt They won the battle at the cost of many lives. Free movement within the European Union must not be implemented at the cost of the EU's erecting barriers against the surrounding world.

at cost of

16,710,000 results on the web

at the cost of

1295,000,000 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 it can't be at the cost of the other things we are trying to achieve here.
Once students' rights come at the cost of others, it is my job to intervene.
This option renders a better image at the cost of increased calculation.
However, this must not come at the cost of social uncertainty.
financed public investment will produce a short
Instead, debt
They won the battle at the cost of many lives.
Free movement within the European Union must not be implemented at the cost of the EU's erecting barriers against the surrounding world.