It really depends in which way they complained to Hetzner. Write a stern email? Hetzner could forward it to you and give time to respond. Preliminary injunction (not uncommon in trademark cases), they'd have no choice in that matter.

Generally, I would recommend not hosting APIs on the primary domain for exactly that reason - it's too easy to be hit with some sort of complaint and have that domain cut off (DMCA, preliminary injunction, SPAM complaint against your mail server, shitty host, ...).