The amount of electricity end-users demand from the distribution grid is going to be more uncertain in the future. In this paper, we empirically investigate how variation in demand (which we refer to as demand uncertainty) influence the pricing behavior of distribution system operators (DSOs) that are subject to an ex-ante revenue cap and a penalty scheme that is activated when the cap is violated. Using data from 162 Swedish electricity DSOs observed over 13 years, our econometric investigations shows that demand uncertainty leads to lower electricity distribution prices under the current ex-ante revenue cap regime, but not under the previous ex-post rate-of-return regime. We also find that DSOs react more strongly to uncertainty when they have a relatively higher debt ratio. While lower network prices may sound like a good outcome for customers, it also leads to reduced revenue and a slowdown in investments. Thus, when society benefits from distribution network expansions and the DSOs are regulated by ex-ante revenue caps, the regulator should consider regulating the debt ratio.