In my app, I simply switched to an AWS http check and set up a "Heartbeat" controller with an Index action; there was no logic, but it returned a 200 when AWS hit /Heartbeat. Your IIS log will catch this request by default; your application infrastructure wasn't outlined in your question, but in my case the lack of any logic in the action was sufficient.
However, implementing your own IHttpModule is another option here. You're at the raw HttpApplication level here, so you don't tap into the MVC pipeline at this point. I've used it for (among other things), forcing the https redirect in my AWS-hosted apps by looking for the "X-Forwarded-Proto" header.