A gas distribution network connects consumers to a source in gas. It is managed by a network operator, whose task incurs various costs, some of which may not be attributable to a particular consumer. Assuming that the operator wishes to recover these costs by charging for its services, the problem is then to determine how much each consumer should pay. In other words, how should these costs be shared among consumers. In this paper, we address this problem and propose cost sharing rules that depend on the network and the demands of the consumers. To that end, we adopt a normative approach and resort to three principles: (i) the independence of higher demands principle, (ii) the connection principle and (iii) the uniformity principle. Applying (i) and (ii), we derive the Connection rule and applying (i) and (iii), we derive the Uniform rule. It appears that (ii) and (iii) are incompatible. In order to make a trade-off between these two principles, we propose the Mixed rules, which compromise between the Connection rule and the Uniform rule. For each cost sharing rule, an axiomatic characterization is provided. Then, we show that the Connection rule coincides with the multi-choice Shapley value of a specific multi-choice game derived from the network and the demands of the consumers. Moreover, the Connection rule is in the Core of this specific multi-choice game. Similarly, we show that the Uniform rule coincides with the multi-choice Equal division value and the Mixed rules coincide with the multi-choice Egalitarian Shapley values.