Multi-agents cost-sharing games are commonly used for modeling settings in which different entities share resources. For example, the setting in which entities need to route messages in a network is modeled by a network-formation game: the network is modeled by a graph, and each agent has to select a path satisfying his reachability objective. In practice, the objectives of the entities are often more involved than reachability. The need to specify and reason about rich specifications has been extensively studied in the context of verification and synthesis of reactive systems. This paper suggests and analyzes a generalization of cost-sharing games that captures such rich specifications. In particular, we study network-formation games with regular objectives. In these games, the edges of the graph are labeled by alphabet letters and the objective of each player is a regular language over the alphabet of labels. Thus, beyond reachability, a player may restrict attention to paths that satisfy certain properties, referring, for example, to the providers of the traversed edges, the actions associated with them, their quality of service, or security. Our results show that the transition to regular objectives makes the game considerably less stable.