In astronomy the study of variable stars that is, stars characterized by showing significant variation in their brightness over time has made crucial contributions to our understanding of many phenomena, from stellar birth and evolution to the calibration of the extragalactic distance scale. In this article, we develop a method for analyzing multiple, (pseudo)-periodic time series with the goal of detecting temporal trends in their periods. We allow for nonstationary noise and for clustering among the various time series. We apply this method to the long-standing astronomical problem of identifying variable stars whose regular brightness fluctuations have periods that change over time. The results of our analysis show that such changes can be substantial, raising the possibility that astronomers' estimates of galactic distances can be refined. Two significant contributions of our approach, relative to existing methods for this problem, are as follows: 1. The method is nonparametric, making minimal assumptions about both the temporal trends themselves but also the covariance structure of the nonstationary noise. 2. Our proposed test has higher power than existing methods. The test is based on inference for a high-dimensional normal mean, with control of the false discovery rate to account for multiplicity. We present theory and simulations to demonstrate the performance of our method. We also analyze data from the American Association of Variable Star Observers and find a monotone relationship between mean period and strength of trend similar to that identified by Hart, Koen, and Lombard (2007).