I read the customer reviews on Amazon: 3.6 rating out of 5.
One reviewer noted it doesn't work with Fire TV. Another noted it doesn't work with any device connected to a TV. After a software update, there are mixed results with any type of Amazon tablet.
I'll assume the Contour app is programmed to NOT work on a TV. If it did work on a TV, that'd be one less cable box you'd rent from Cox. Decipher that as you want.
I don't have Fire TV, but I know its operating system (Fire) is based on Android. Android is based on Linux. Meaning, Fire is open-source software. Open-source software means a programmer can modify and distribute the software free-of-restrictions and free of any licensing fees. Some people will call this "hacked" or "jail broke" but it's not...it's completely legal.
When a programmer modifies an Android-based program, they'll distribute the modification as an APK file (Android application PacKage). You can find an APK version of this app that's compatible with the Fire TV Stick. Search: Firestick APK Contour