For evolving datasets with continual reports, the composition rule for differential privacy (DP) dictates that the scale of DP noise must grow linearly with the number of the queries, or that the privacy budget must be split equally between all the queries, so that the privacy budget across all the queries remains bounded and consistent with the privacy guarantees. To avoid this drawback of DP, we consider datasets containing almost periodic time series, composed of periodic components and noisy variations on top that are independent across periods. Our interest in these datasets is motivated by that, for reporting on private periodic time series, we do not need to divide the privacy budget across the entire, possibly infinite, horizon. Instead, for periodic time series, we generate DP reports for the first period and report the same DP reports periodically. In practice, however, exactly periodic time series do not exist as the data always contains small variations due to random or uncertain events. For instance, the energy consumption of a household may repeat the same daily pattern with slight variations due to minor changes to the habits of the individuals. The underlying periodic pattern is a function of the private information of the households. It might be desired to protect the privacy of households by not leaking information about the recurring patterns while the individual daily variations are almost noise-like with little to no privacy concerns (depending on the situation). Motivated by this, we define DP for almost periodic datasets and develop a Laplace mechanism for responding to linear queries. We provide statistical tools for testing the validity of almost periodicity assumption. We use multiple energy datasets containing smart-meter measurements of households to validate almost periodicity assumption. We generate DP aggregate reports and investigate their utility.