Most of the template formats need a cache folder to keep their compiled files so that they run efficiently
Since RC5 we create a subfolder for the template type used. Even though php templates does not need compilation Restler attempts creating php folder under the default cache location, which is cache folder located in the same folder as the index.php
In your case it is
/home/xxxxxxx/public_html/concrete/api/cache
You should update the cache folder to keep it outside the web root by adding
Defaults::$cacheDirectory = '/home/xxxxxxx/cache';
And then make sure the cache folder is writable
Then HtmlFormat will create the php/twig/blade folder depending on your template preference and add the compiled files inside