Online pricing mechanisms have been widely applied to resource allocation in multi-agent systems. However, most of the existing online pricing mechanisms assume buyers have fixed valuations over the time horizon, which cannot capture the dynamic nature of valuation in emerging applications. In this paper, we study the problem of revenue maximization in online auctions with unknown time discounting valuations, and model it as non-stationary multi-armed bandit optimization. We design an online pricing mechanism, namely Biased-UCB, based on unique features of the discounting valuations. We use competitive analysis to theoretically evaluate the performance guarantee of our pricing mechanism, and derive the competitive ratio. Numerical results show that our design achieves good performance in terms of revenue maximization on a real-world bidding dataset.