It might not be the app's fault (for once).
Your device is probably utilizing its power saving features and suspending the app while the screen is off.
Different brands of devices have different ways of accessing the menu where you can add an app to the exceptions list, but on Samsung devices it's as follows:
Long press the Roku app icon and then select app info.
From there, press the battery heading.
There will be three options, with one of them being "Unrestricted". This is the one you want.