köken: From a (“at”) + todo (“every”) + custo (“cost, price”), perhaps a calque of English at all costs (literally “at all conditions”, “no matter the possibilities”), where cost refers not to price but to the obsolete sense of “manner”, “option”, “possibility”, being related to choice and (distantly, via Proto-Indo-European) also to Portuguese gosto.
Literally, “it may cost what it will”. Probably after French coûte que coûte. Compare the variant above, which is equivalent to the French variant quoi qu'il (en) coûte, further the early attestation below (in a translation from French), and finally Dutch koste wat kost, which last is clearly a loan translation.