What is the status of a kosher pot that was used for cooking (with kosher ingredients) on Shabbat? Is it now not kosher anymore? How can I kasher it? For usual kasherin there seem to be two rules- one with cleaning, heating and so on, the other being that if something wasn’t used for a while. Could you explain when what applies or if both are equal?

The Mishnah Brurah (318; 4) quotes the Magen Avraham which learns from the Rashba that a vessel used to cook food on Shabbat in a prohibited way can’t be used unless koshered. There are lenient opinions that dispute the Magen Avraham’s conclusion and allow the vessel to be used without any further procedure (See Yalkut Yosef Shabbat 3, p.27-28) A metal pot should be koshered by dismantling that which can, thorough cleaning, waiting 24 hours from last usage, immersing in bubbling boiling water for a few seconds and washing with cold water; or blow torching.