Does it hurt web rankings when the site is hosted on google app engine as a subdomain of appspot.com?

[Jason]

It's such a bad idea to have someone else's domain as an address of your website.

It influences your rankings in a major way, but not in a way you think, not directly.
When your site becomes successful, or you need something google app engine doesn't support, or you want to change your hosting or you just want to finally get your own domain name, THEN all the hard work you've spent months or years building your website's ranking will be lost.

So basically, in the short term (shallow thinking) it doesn't influence your rankings. When you think things through you realize how bad idea it is. Don't let the 3rd party control your website. Just get your own domain name. It's like $5-$10/year.