Anomaly detection in real-world time-series data is a challenging task due to the complex and nonlinear temporal dynamics involved. This paper introduces KoopAGRU, a new deep learning model designed to tackle this problem by combining Fast Fourier Transform (FFT), Deep Dynamic Mode Decomposition (DeepDMD), and Koopman theory. FFT allows KoopAGRU to decompose temporal data into time-variant and time-invariant components providing precise modeling of complex patterns. To better control these two components, KoopAGRU utilizes Gate Recurrent Unit (GRU) encoders to learn Koopman observables, enhancing the detection capability across multiple temporal scales. KoopAGRU is trained in a single process and offers fast inference times. Extensive tests on various benchmark datasets show that KoopAGRU outperforms other leading methods, achieving a new average F1-score of 90.88\% on the well-known anomalies detection task of times series datasets, and proves to be efficient and reliable in detecting anomalies in real-world scenarios.