Do AWS backups of large empty volumes take longer than small ones? AWS provides backup functionality for RDS databases and snapshots functionality for EBS volumes.

Given a small amount of data, would the backup and restore time change depending on whether it's stored on a large disk or a small one?

In case you are asking yourself why one would put a small amount of data on a large disk: EBS gp2 volumes offer more IOPS (IO operations) with more storage (3 IOPS per 1 GB). So if you need a disk with plenty of IO operations but little persistent data, you might end up with a few megs of data sitting on a 1000 GB disk.

If my assumption that backups might take longer for larger disks is true, then it might be worth deciding for the more expensive io1 disk type instead, which can scale with up to 50 IOPS per GB.
Firstly I would argue that backup time does not matter,  as it usually doesn't correlate with the restore time. 

In regards to restore time,  snapshots restore on volumes almost instantly,  as data is only copied onto the volume from S3 when requested. 

So as long as you only request actually used data from the volume, it would only matter how much data is used on the volume to determine the time to restore. 

This link explains the process well https://docs.aws.amazon.com/AWSEC2/latest/UserGuide/ebs-restoring-volume.html