Drug addiction (DA) is a long-term and relapsing brain disorder with limited effective treatments. Electroencephalography (EEG) is a highly promising tool for investigating DA. This letter proposes an effective sequential convolutional long short-term memory (LSTM) network based on temporal-spatial conversion for DA detection from EEG signals. First, the multi-channel EEG time series are converted into a few EEG topomaps composed of RGB colors, to reduce the temporal-spatial redundancy of EEG signals. Then these EEG topomaps are input to the convolutional module to extract the spatial features of brain activity under DA condition. Next, considering the EEG temporal correlation, an LSTM module is introduced to adaptively capture significant sequential information like time series. Meanwhile, a contrastive loss function is defined for reinforcing the temporal-spatial features, to improve DA detection. Experiments on the DA dataset show that the proposed network is simple and universal, and can achieve better detection performance compared to several existing approaches.