How can i download Lightroom on desktop without paying for one when I already have an account

Can you clarify what type of account you have on your phone? If you mean you have the free version of the Lightroom app, that free app is only available on the phone. To use the Lightroom app on your Mac will require a full subscription service. There is/was a lower grade subscription service available on the phone, if that's what you have you would need to upgrade it to the full subscription in order to use the app on your Mac.