You can actually use Pingdom to check for this kind of error, rather than a straight forward check for a response from the server you can configure the check to look for a particular string on the page. For instance, if your page has some kind of consistent bit of text on the homepage then you can look for that.
This Pingdom post has some info about checking for strings:
Make the most of your HTTP check: best practice for optional settings
You can also check that it doesn’t have particular string, so you should be able to make it flag it as down if it shows the text “Error establishing a database connection”.