The effective distribution of relief to an emergency logistics system plays a crucial role during the disaster response phase. Considering stochastic characteristics of relief demand, this study investigates the robust optimization of a multi-objective multi-period location-routing problem for epidemic logistics, a special emergency logistics, with uncertain scenarios. A corresponding robust multi-objective multi-period optimization model is proposed, which aims to determine the optimal location of temporary relief distribution centres and route planning simultaneously. The optimization objectives include the total travel time, the total cost, and the disutility of relief service. To solve the above optimization model, a preference-inspired co-evolutionary algorithm with Tchebycheff decomposition (PICEA-g-td) is given. The performance of the proposed PICEA-g-td is evaluated by comparing it with NSGA-II, MOEA/D and PICEA-g. The experimental results show that the proposed algorithm performs better than the other three algorithms in terms of the solution quality. Finally, some useful management insights are obtained.