PS3 error 80710723 mean a problem in the network / router. This error usually happens on upgrades and many are going to update this with 1192 MB of Gran Turismo 5 (although it can also happen with other games), here the solution:
Within the XMB menu system enters PS3 SETTINGS> NETWORK SETTINGS> SETTINGS INTERNET
1) Wired: CABLE> CUSTOM> MANUALLY ENTER> AUTOMATICALLY DETECT> AUTOMATIC> NO SET> MANUAL (DNS1: 208.67.222.222 - DNS2: 208.67.220.220)> AUTOMATIC> NO USE> ON> BUTTON X> X BUTTON
2) Wi-Fi: Wireless> Enter MAUNALMENTE> (PLACE OR CONTINUE SSID)> (INSERT PASSWORD OR GO)> CUSTOM> AUTOMATIC> NO SET> MANUAL (DNS1: 208.67.222.222 - DNS2: 208.67.220.220)> AUTOMATIC> NO USE> ON> BUTTON X> X BUTTON
Remember that steps should be followed in the correct order and literally. I hope you have served. Greetings.