You have to wonder why it's so hard to put that on their 503 error page. I suspect something's much more broken than they're letting on.
Example of a response I see:
< x-cache: Error from cloudfront < via: 1.1 bdf85d6d4811ab08c57841855a848f8a.cloudfront.net (CloudFront) < x-amz-cf-pop: LAX54-P11 < x-amz-cf-id: nTQ-y1Ut3F-04jUCDM09ordCtj0CMkVmmtZTe__BtzEr1sMJu7rKaw== < age: 76773