Problem

How to use an API key (Outbound authentication) with a custom routing policy in API Manager.

Resolution

Outbound authentication is not applied when a custom routing policy is selected.

One workaround for this would be to add apikey in the custom routing policy.
For example you could use External connections --> Client Credentials--> API Keys.
And then use this API key in the 'Connect to url' filter (authentication tab).
Or you could use the "Add HTTP header" filter to send in the API key as a header.