At any cost

the price paid to acquire, produce, accomplish, or maintain anything:
the high cost of a good meal.
an outlay or expenditure of money, time, labor, trouble, etc.:
what will the cost be to me?
a sacrifice, loss, or penalty:
to work at the cost of one’s health.
costs, law.

money allowed to a successful party in a lawsuit in compensation for legal expenses incurred, chargeable to the unsuccessful party.
money due to a court or one of its officers for services in a cause.

to require the payment of (money or something else of value) in an exchange:
that camera cost $200.
to result in or entail the loss of:
carelessness costs lives.
to cause to lose or suffer:
the accident cost her a broken leg.
to entail (effort or inconvenience):
courtesy costs little.
to cause to pay or sacrifice:
that request will cost us two weeks’ extra work.
to estimate or determine the cost of (manufactured articles, new processes, etc.):
we have costed the manufacture of each item.
to estimate or determine costs, as of manufacturing something.
cost out, to calculate the cost of (a project, product, etc.) in advance:
the firm that hired him just costed out a major construction project last month.
at all costs, regardless of the effort involved; by any means necessary:
the stolen painting must be recovered at all costs.
also, at any cost.
noun
the price paid or required for acquiring, producing, or maintaining something, usually measured in money, time, or energy; expense or expenditure; outlay
suffering or sacrifice; loss; penalty: count the cost to your health, i know to my cost

the amount paid for a commodity by its seller: to sell at cost
(as modifier): the cost price

(pl) (law) the expenses of judicial proceedings
at any cost, at all costs, regardless of cost or sacrifice involved
at the cost of, at the expense of losing
verb costs, costing, cost
(transitive) to be obtained or obtainable in exchange for (money or something equivalent); be priced at: the ride cost one pound
to cause or require the expenditure, loss, or sacrifice (of): the accident cost him dearly
to estimate the cost of (a product, process, etc) for the purposes of pricing, budgeting, control, etc
n.

c.1200, from old french cost (12c., modern french coût) “cost, outlay, expenditure; hardship, trouble,” from vulgar latin -costare, from latin constare, literally “to stand at” (or with), with a wide range of figurative senses including “to cost.” the idiom is the same one used in modern english when someone says something “stands at x dollars” to mean it sells for x dollars. the latin word is from com- “with” (see com-) + stare “to stand,” from pie root -sta- “to stand” (see stet).
v.

late 14c., from old french coster (modern french coûter) “to cost,” from cost (see cost (n.)).
also, at any price . see at all costs
see:

arm and a leg, cost an
at all costs
pretty penny, cost a