This is a confluence of different things, so this is gonna be a kind of complicated answer.
While we're working on getting https everywhere, SO is available over https but it's not really supported yet. Because Googlebot will, at least in our experience, prefer crawling over https we explicitly redirect request from googlebot ips to http. We'll probably stop doing this once our infrastructure is ready for https everywhere.
The API wants some requests to come over https but is ok with http for all unauthenticated traffic. With regards to the API, https everywhere is already good to go. We don't require it except for authenticated requests, because that would be a breaking change in the 2.x-family.
Mix the two, and API requests that require authentication that also come from known googlebot will be redirected to http and then fail. Woops.
We're looking at fixing this, there's no reason not to allow googlebot to index (or otherwise query)
I'll update this answer once the fix is in.
Our routing was updated last night to fix this, no traffic to
api.stackexchange.com should be redirected from https to http now; regardless of origin.