As long as your new device is on the same platform as your previous phone (e.g. iPhone and iPad both run iOS) and you are signed in to the appropriate app store with the same ID, you will be able to download any previously purchased apps to any new devices. The app is not transferable to a new operating system, so if you switch from iPhone to Android, for example, your previous purchases in the Apple App store will not be available for your new Android phone, or vice versa. In that case, unfortunately, you would need to pay for the app again.
If you are trying to reinstall the original V1 version of the app (before 2018), it is no longer maintained and no longer listed in the app store, but if you had purchased it and want to restore it on a current phone the instructions are here:
And if none of that works you can send me an email through the “contact” link in the header.