Despite OnlineOrNot being the nice kind of bot, it may occassionally report false-positives (tells you the site is down, when it's actually up) when used on websites that use Cloudflare.
In Cloudflare, open the dashboard for your website, and navigate to "Firewall" in the top menu.
Once in the Firewall, add a new filter of "User agent", "contains", and "OnlineOrNot", as shown in the screenshot below.
OnlineOrNot may be tripped up by Cloudflare's Bot Fight Mode, if your website has it enabled. When this happens, OnlineOrNot starts reporting your page as DOWN.
If you are a user of Cloudflare's Pro plan, it is possible to customize Bot Fight Mode and add an exception for OnlineOrNot. If you're using Cloudflare's free tier, unfortunately the only workaround is to disable Bot Fight Mode.