To fully exploit the rich state and fault information embedded in the acoustic signals of a hydraulic plunger pump, this paper proposes an intelligent diagnostic method based on sound signal analysis. First, acoustic signals were collected under normal and various fault conditions. Then, four distinct acoustic features—Mel Frequency Cepstral Coefficients (MFCCs), Inverse Mel Frequency Cepstral Coefficients (IMFCCs), Gammatone Frequency Cepstral Coefficients (GFCCs), and Linear Prediction Cepstral Coefficients (LPCCs)—were extracted and integrated into a novel hybrid cepstral feature called MIGLCCs. This fusion enhances the model’s ability to distinguish both high- and low-frequency characteristics, resist noise interference, and capture resonance peaks, achieving a complementary advantage. Finally, the MIGLCC feature set was input into a double layer long short-term memory (DLSTM) network to enable intelligent recognition of the hydraulic plunger pump’s operational states. The results indicate that the MIGLCC-DLSTM method achieved a diagnostic accuracy of 99.41% under test conditions. Validation on the CWRU bearing dataset and operational data from a high-pressure servo motor in a turbine system yielded overall recognition accuracies of 99.64% and 98.07%, respectively, demonstrating the robustness and broad application potential of the MIGLCC-DLSTM method.