articles - At the cost of about or At a cost of about

According to preliminary estimates, the project should be completed by 2020 at a cost of about RUB 40 bn.

or

According to preliminary estimates, the project should be completed by 2020 at the cost of about RUB 40 bn.

Are both options okay, or is the first better?

We are introducing the sum to the reader, so logically it should be the first option. But there's a lot of google-hits for "at the cost of" too.

Answer

I'd say either is okay. In a lot of cases, a and the are more or less interchangeable, and I'd say this is one of these.

I believe "a" is technically correct, because (technical side-note, this is called antecedent basis in the patent world) before you introduce the idea of the cost, saying "the" is more or less meaningless.

It would be kind of like talking to a friend and introducing the idea that I want to buy a specific car by saying out of nowhere, "I want to buy that car." Well, great, but it sure helps if we can introduce the meaning of "that" before I declare such a fact.

The same is true here, but to a lesser extent.

But again, either is perfectly acceptable in most cases. There might be some rule of grammar that more specifically pinpoints a reason to choose one over the other, but as a native speaker and not an English major, I'd say that it really doesn't matter which you choose. Honestly, "the" sounds a little more comfortable to me, so I'd probably go with that.