Okay, some days ago I figured out what caused the problem. The type of problem is kinda embarrasing but I want to share the solution if somebody else stumbles over this.
I replaced the the domains from local to staging environment with the protocol (http:// or https://). But I did not think about that there could be some more database entries without the protocol.
So after I run search and replace there were still some references to the local environment.
The (obvious) solution: If the protocol doesn’t change from the two environments just replace the domains without it.
As I wrote, this is kinda embarrasing because it’s such a obvious problem. But it happens. 