What is page_fast_cache? The Cache Router module advertises that it "has an option to utilize the page_fast_cache part of Drupal in order to reduce the amount of resources needed for serving pages to anonymous users."

However, the only Google hits for "drupal page_fast_cache" point back to cacherouter.

What is page_fast_cache?
I believe the project page is referring to page_cache_fastpath(), which is called during Drupal bootstrap from the following code:

if (variable_get('page_cache_fastpath', FALSE) &amp;&amp; page_cache_fastpath()) {
  exit;
}


The documentation page reports the following description for the function:


  By implementing page_cache_fastpath(), a special cache handler can skip most of the bootstrap process, including the database connection. This function is invoked during DRUPAL_BOOTSTRAP_EARLY_PAGE_CACHE.


The example code reported in that page helps to understand how the function needs to be implemented, and what should do.

function page_cache_fastpath() {
  $page = mycache_fetch($base_root . request_uri(), 'cache_page');
  if (!empty($page)) {
    drupal_page_header();
    print $page;
    return TRUE;
  }
}


If the module reports to utilize it, it means it actually defines the page_cache_fastpath() function, which is called by Drupal core code.
To notice that it's not a hook, but a function. This means that a module called "mycache" doesn't need to implement mycache_page_cache_fastpath(), but page_cache_fastpath(). In fact, the bootstrap code is explicitly calling page_cache_fastpath(), as it is evident in the code I reported at the beginning of this answer.