Cardiac CT plays an important role in diagnosing heart diseases but is conventionally limited by its complex workflow that requires dedicated phase and bolus tracking [e.g., electrocardiogram (ECG) gating]. This work reports initial progress towards robust and autonomous cardiac CT exams through deep learning (DL) analysis of pulsed-mode projections (PMPs). To this end, cardiac phase and its uncertainty were simultaneously estimated using a novel projection domain cardiac phase estimation network (PhaseNet), which utilizes a sliding-window multi-channel feature extraction approach and a long short-term memory (LSTM) block to extract temporal correlation between time-distributed PMPs. Monte-Carlo dropout layers were utilized to predict the uncertainty of deep learning-based cardiac phase prediction. The performance of the proposed phase estimation pipeline was evaluated using accurate physics-based emulated data.