Time Capsule first backup is slow (like a week to complete)

After about six hours of waiting for Godot, I tried this experiment: put the Mac to sleep (power not connected) and see if (still sloooow) backup proceeds while sleeping. It did not. (I let it sleep for a half hour, so the presence or absence of progress would be more apparent.) But... on re-awakening, the backup was more than 10x faster.

Also, I found that when power is connected, backup does continue while sleeping. (PowerNap feature, checkbox in System Preferences > Energy Saver panel.) So while both laptop and I were asleep, 90% of the backup completed.

Solution? I don't know how general it is, and this is post-hoc reasoning, but my suggestion is: try putting your Mac to sleep for a while, with power disconnected. On awakening, it might just be 10x faster.