Detection of cyber attacks in smart power distribution grids with unbalanced configurations poses challenges due to the inherent nonlinear nature of these uncertain and stochastic systems. It originates from the intermittent characteristics of the distributed energy resources (DERs) generation and load variations. Moreover, the unknown behavior of cyber attacks, especially false data injection attacks (FDIAs) in the distribution grids with complex temporal correlations and the limited amount of labeled data increases the vulnerability of the grids and imposes a high risk in the secure and reliable operation of the grids. To address these challenges, this paper proposes an unsupervised adversarial autoencoder (AAE) model to detect FDIAs in unbalanced power distribution grids integrated with DERs, i.e., PV systems and wind generation. The proposed method utilizes long short-term memory (LSTM) in the structure of the autoencoder to capture the temporal dependencies in the time-series measurements and leverages the power of generative adversarial networks (GANs) for better reconstruction of the input data. The advantage of the proposed data-driven model is that it can detect anomalous points for the system operation without reliance on abstract models or mathematical representations. To evaluate the efficacy of the approach, it is tested on IEEE 13-bus and 123-bus systems with historical meteorological data (wind speed, ambient temperature, and solar irradiance) as well as historical real-world load data under three types of data falsification functions. The comparison of the detection results of the proposed model with other unsupervised learning methods verifies its superior performance in detecting cyber attacks in unbalanced power distribution grids.