We introduce a new multiple change point detection procedure, called the Functional Block Cumulative Sum (FB-CUSUM), for detecting changes in the mean, variance, and shape of a functional process. Our method uses total variation denoising and a new region-merging procedure to split projections of a functional time series into overlapping blocks that likely contain at most one change point each. The univariate CUSUM statistic can then be applied blockwise to find all change points for each projection. We establish the statistical properties of our procedure and show that its computational cost grows linearly in the number of observations. Numerical simulations show that our method maintains the nominal false discovery rate, accurately detects the number and locations of change points under a variety of scenarios, and is robust compared to existing methods. Finally, we apply our method to a large time series of water vapor mixing ratio profiles from Atmospheric Emitted Radiance Interferometer measurements.