The error code 80710723 on the Sony PlayStation 3 video game console means there is a network or router error with the device. This error might occur while trying to install an update to the PS3. When this error occurs, it is possible to try to restart the update to correct the error.
If restarting the update does not work, the error might be caused by other issues. One such issue is that the network is simply too congested to allow you to do the update. Another issue is that the console is overheating. In these cases, waiting until later to do the update might be advisable. If that does not work, it might be necessary to alter the wireless connection settings on the console.