Running PHP without extension without using mod_rewrite?

An alternative is to use content negotiation. Turn on multiviews:

Options +MultiViews

If a named resource doesn't exist, Apache will glob for the file, then sort based on the media type and content encoding requirements send by the browser. If there's only one file (your PHP script), then that's what the URL resolves to.

You could also force the mime type of a specific file in your .htaccess:

<Files app>
ForceType application/x-httpd-php
</Files>