It would cost in the same sense that it costs to lower the tax rate: a lower amount of future payments to the federal government from that particular revenue source.
Of course, a lot of student loan debt will never be paid anyway for various reasons, such as deferment, default, or death of the borrower.
Loan forgiveness won't result in any immediate payments from the federal government, because these loans are held by the federal government rather than by banks, something that a lot of people seem to misunderstand. The payments were already made years ago when the loans were disbursed.
There would be no payments from the treasury so no direct cost for the government. The "cost" would be an opportunity cost of collecting payments. Which from a government perspective is offset by the spending and savings in the broader economy for not collecting those payments.
Forgive, not pay for.
> Legality still pending AFAIK.
Oral arguments were heard by the Supreme Court a few weeks ago.
> the debt cannot be dispelled through bankruptcy
It can, actually, though it's difficult.