The adept godling archetype weapon champion extraordinary ability favored weapons says, "If the weapon champion has a feat that applies to one of these weapons (such as Weapon Focus), it applies to all the weapons in the group" (emphasis mine).
So, while the extraordinary ability weapon advantage does allow the adept godling weapon champion to "[a]dd one weapon to the weapon champion’s list of favored weapons"—like adding to the axes weapon group the heavy crossbow—, adding this weapon does not allow the champion to suddenly use feats in a nonstandard way. For example, an adept godling weapon champion like the one described previously cannot, for example, employ the feat Power Attack when firing a heavy crossbow!
That's because the feat Power Attack and the vast majority of other combat feats do not apply to one weapon exclusively—like, for example, the feats Improved Critical, Weapon Focus, and Weapon Specialization do—, and, instead, can apply to any appropriate weapon the champion happens to be using.
Thus, for example, a champion that picks the axes weapon group and adds to it the heavy crossbow and possesses the feats Improved Critical (handaxe) and Weapon Focus (handaxe) will realize the benefits of those two feats with all weapons in the axes group and with the heavy crossbow, but his Power Attack feat remains useless when firing his heavy crossbow.
However, third-party material can't anticipate what Paizo and other publishers will create, making third-party material easiest to design when its focused on the core rules exclusively. That is, for example, what happens when a weapon champion that possesses a feat like Improved Whip Mastery and that adds the whip to the axes weapon group is a mystery.
With that in mind, this GM recommends that only feats that specify a creature can pick any weapon—like the feat Weapon Focus that's mentioned in the description of the extraordinary ability favored weapons—apply their benefit to all weapons in the favored weapon group and, by extension, any weapons added afterward by the extraordinary ability weapon advantage. This is not because this GM hates fun! It's because adjudicating how weapon-specific feats like Gun Twirling, Master Siege Engineer, or Shaft and Shield apply to weapons they aren't designed to affect is an exercise in futility and mind-reading and may generate vicious arguments between the player who reads a feat one way and the GM who reads it another.