Image: Google
You don't have to worry about losing your Google Authenticator one-time codes. An update for the Authenticator on Android and iOS now stores the backups of your codes in your Google account. This means, you don't need to reauthorize all your linked apps or scan a QR code when you get a new phone.
When you have the latest app version, you must follow the prompts to sign into Google and enable syncing. Of course, make sure your Google account is secure to prevent intruders from misusing the Authenticator.