Here's a nonextensive list of open-world games vaguely like TDU that are fun enough to hopefully occupy us. Feel free to add to.
Forza Horizon Demo (duh)
NFS Hot Pursuit (2010)
GTA IV
Just Cause 2
Mafia II
Driver: San Francisco
Fallout: New Vegas (GOTY Ed. would easily take the entire month)
Fallout 3
Red Dead Redemption
The Elder Scrolls V: Skyrim
ArmA 2
ArmA 3 (s**n)
Deus Ex: HR
Borderlands (free on PSN+ as of last week)
Borderlands 2
Minecraft
SW:TOR