Each of the seven deadly sins has different ways of generating demonic energy, influencing the people around them, and a different kind of connection with their hosts.
Lust influences people to desire all kinds of pleasure, especially sexual, and she enjoys that pleasure mostly through Lucien and his women.
Wrath influences every feeling of anger in people around her and enjoys those feelings mainly through Donna and the people who interact with her, especially those she enrages and the ones she unleashes fury on.
In the same way, Gluttony works. Her special ability is to devour any type of energy and turn it into demonic energy. She can do that by absorbing any energy directly, but the most efficient ways are by eating or drinking delicious foods that contain powerful energies, and she cannot do it personally.