Should I backup a database with only temporary data?
If there is a database with data on it, someone, at some point, will consider it important. So best approach would be run a full backup to avoid any hassle. Maybe weekly with couple of daily differentials. If there are frequent changes then maybe a daily full backup and one differential in the middle of the day.

If the data is temporary and can be recreated somehow, then I would configure the recovery model to SIMPLE and take a FULL backup daily. And to avoid spaces issues, if the data is not so important and can be recreated afterwards, I would add a process to empty all tables before running the backup. Then configure another step after backup process in order to add fresh data once again so anyone using the database has data to continue working.

And last, but not least, you could just script the database and recreate it from those scripts. Do you have your database code under source control? That would help a lot on this case. We have couple of databases with similar characteristics and we only save the recreation scripts, data can be easily recreated with couple of scripts.