If you already have the app installed on a device currently, it should replicate to the new device, assuming its the same Spectrum and Roku account.
However, I have seen reports from some saying this didnt happen for them, that the Spectrum app did not replicate to their new device on the same Spectrum & Roku account.
You might be able to call Spectrum in such a scenario and they may give you a link to a private version of the app with a code, but there are also mixed reports for that as well.
Ultimately the best (and only) way to find out/know for sure is to buy and setup a new device.