Unfortunate as it is, Call of Duty: Black Ops Cold War will not offer any free next-generation upgrades for players.
The new Call of Duty: Black Ops Cold War will see a cross-generation release but without any plans to offer free next-generation upgrades. According to a leak (via Charlie Intel) from earlier today, Black Ops Cold War will be available for purchase in three different editions. The standard edition will be the base game for PlayStation 4, Xbox One, and PC. The cross-generation edition though will bundle the base game with the next-generation versions. Hence, the only way to play Black Ops Cold War on PlayStation 5 and Xbox Series X will be to purchase the cross-generation edition. There will be no free upgrades for owners of the standard edition.
