I also use Hostgator for my website (www.texbiker.net). In the last 2 or 3 weeks I had a similar problem. It got to a point that Hostgator called my phone as well as sent me an email. Recently I also started getting some 500 internal server errors. At the time I was running WP 3.5.1. I am not sure what I did, if anything, but the problem went away. I did clean out the posts revisions (over 9,000) and check some of my plugins.

With the number of calls to the login and index php files I would guess some bot was trying to exploit the login weakness where people do not use strong passwords and the username Admin. The reason I think this is the index.php page has the link to the login link. After failing it went away.

A problem I am working on now is the PHP memory limit Hostgator imposes on shared hosting accounts. One plugin (All-in-One Event Calendar) causes the problem: Allowed memory size of 268435456 bytes exhausted /home/texbiker/public_html/blog/wp-includes/wp-db.php
This only happens when I try to list all of the events I have created over the years.

Glad that you are back at WP Tavern. Keep writing.