Online identification of postcontingency transient stability is essential in power system control, as it facilitates the grid operator to decide and coordinate system failure correction control actions. Utilizing machine learning methods with synchrophasor measurements for transient stability assessment has received much attention recently with the gradual deployment of wide-area protection and control systems. In this paper, we develop a transient stability assessment system based on the long short-term memory network. By proposing a temporal self-adaptive scheme, our proposed system aims to balance the trade-off between assessment accuracy and response time, both of which may be crucial in real-world scenarios. Compared with previous work, the most significant enhancement is that our system learns from the temporal data dependencies of the input data, which contributes to better assessment accuracy. In addition, the model structure of our system is relatively less complex, speeding up the model training process. Case studies on three power systems demonstrate the efficacy of the proposed transient stability as sessment system.