This app was mentioned in 12 comments, with an average of 2.58 upvotes
You also get the LastPass Authenticator as well so for that $12 a year things are pretty locked down. I know OP's looking for free but I hope he reads these posts and considers it.
Yes, turning them off would be the recommend way if using Google Authenticator. But other apps do have a backup system like you are talking about, such as LastPass Authenticator or Authy
If you use Lastpass, I'd suggest you use Lastpass Authenticator. It backs up your tokens to your vault, so your codes are still there when you reinstall/change devices.
Unfortunately, there's no way to restore your codes in Google Authenticator. You'll have to use recovery options (like printed backup/recovery codes) until you can get a new authenticator app set up.
My Lastpass Authenticator app lets me approve via my watch. But the only thing that I use LP Authenticator for is to get into LP itself. For allo other 2FA, I use Google Authenticator, so I can't promise it would work for your needs. However, it's supposed to be a general purpose 2FA app.
"LastPass Authenticator offers effortless two-factor authentication for your LastPass account and other supported apps." https://play.google.com/store/apps/details?id=com.lastpass.authenticator
Not the extension.... You need Lastpass Authenticator App for IOS or Android. It is the same as Google Authenticator app. Difference is that you can set up PIN or fingerprint.
Android: https://play.google.com/store/apps/details?id=com.lastpass.authenticator&hl=en
IOS: https://itunes.apple.com/us/app/lastpass-authenticator/id1079110004?mt=8
use Lastpass authenticator it has backup option if you have LastPass account
Give it a try :https://play.google.com/store/apps/details?id=com.lastpass.authenticator&hl=en
Free with sync. My goto authenticator currently, as I also use lastPass (free), so it links to lastPass account.
Looks like Lastpass Authenticater does synchronization as well.
"Optional encrypted backup to restore your tokens on a new/reinstalled device"
https://play.google.com/store/apps/details?id=com.lastpass.authenticator
Use Lastpass Authenticator. It syncs between phones and is widely seen as the most secure option.
I use Lastpass but some reviews on the Authenticator are inducing worry https://play.google.com/store/apps/details?id=com.lastpass.authenticator