The noun (Substantiv/Nomen) Mehrkosten is feminine in German. Therefore, the correct definite article in the nominative singular is die. So, you say: die Mehrkosten.
additional costs
This word is composed and its gender is always determined by the root word.
Zusätzliche Kosten, die über die ursprünglich geplanten oder erwarteten Kosten hinausgehen.
Additional costs that exceed the originally planned or expected costs.
Durch die Verzögerung entstanden Mehrkosten. Die Mehrkosten müssen genehmigt werden.
The delay caused additional costs. The additional costs must be approved.