Data dependencies have been extended to graphs to characterize topological and value constraints. Existing data dependencies are defined to capture inconsistencies in static graphs. Nevertheless, inconsistencies may occur over evolving graphs and only for certain time periods. The need for capturing such inconsistencies in temporal graphs is evident in anomaly detection and predictive dynamic network analysis. This paper introduces a class of data dependencies called Temporal Graph Functional Dependencies (TGFDs). TGFDs generalize functional dependencies to temporal graphs as a sequence of graph snapshots that are induced by time intervals, and enforce both topological constraints and attribute value dependencies that must be satisfied by these snapshots. (1) We establish the complexity results for the satisfiability and implication problems of TGFDs. (2) We propose a sound and complete axiomatization system for TGFDs. (3) We also present efficient parallel algorithms to detect inconsistencies in temporal graphs as violations of TGFDs. The algorithm exploits data and temporal locality induced by time intervals, and uses incremental pattern matching and load balancing strategies to enable feasible error detection in large temporal graphs. Using real datasets, we experimentally verify that our algorithms achieve lower runtimes compared to existing baselines, while improving the accuracy over error detection using existing graph data constraints, e.g., GFDs and GTARs with 55% and 74% gain in F1-score, respectively.