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Internet of Things (IoT) sensors in smart buildings are becoming increasingly ubiquitous, making buildings more livable, energy efficient, and sustainable. These devices sense the environment and generate multivariate temporal data of paramount importance for detecting anomalies and improving the prediction of energy usage in smart buildings. However, detecting these anomalies in centralized systems is often plagued by a huge delay in response time. To overcome this issue, we formulate the anomaly detection problem in a federated learning setting by leveraging the multi-task learning paradigm, which aims at solving multiple tasks simultaneously while taking advantage of the similarities and differences across tasks. We propose a novel privacy-by-design federated learning model using a stacked long short-time memory (LSTM) model, and we demonstrate that it is more than twice as fast during training convergence compared to the centralized LSTM. The effectiveness of our federated learning approach is demonstrated on three real-world datasets generated by the IoT production system at General Electric Current smart building, achieving state-of-the-art performance compared to baseline methods in both classification and regression tasks. Our experimental results demonstrate the effectiveness of the proposed framework in reducing the overall training cost without compromising the prediction performance.