If Pokemon League Headquarters is in play and I have Lunatone with the "New Moon" Ability and also Solrock in play, do my Basic Pokemon have an extra {C} cost to their Attacks?
No, but your opponent's Pokemon do.
Source: Obsidian Flames FAQ; TPCi Rules Team (2023-08-10)

Relevant cards

Note: All applicable cards may not be listed below.
SV3_92
SV3_92
SV3_192