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Functional data is a powerful tool for capturing and analyzing complex patterns and relationships in a variety of fields, allowing for more precise modeling, visualization, and decision-making. For example, in healthcare, functional data such as medical images can help doctors make more accurate diagnoses and develop more effective treatment plans. However, understanding the causal relationships between functional predictors and time-to-event outcomes remains a challenge. To address this, we propose a functional causal framework including a functional accelerated failure time (FAFT) model and three causal approaches. The regression adjustment approach is based on conditional FAFT with subsequent confounding marginalization, while the functional-inverse-probability-weighting approach is based on marginal FAFT with well-defined functional propensity scores. The double robust approach combines the strengths of both methods and achieves a balance condition through the weighted residuals between imputed observations and regression adjustment outcomes. Our approach can accurately estimate causality, predict outcomes, and is robust to different censoring rates. We demonstrate the power of our framework with simulations and real-world data from the Alzheimer's Disease Neuroimaging Initiative (ADNI) study. Our findings provide more precise subregions of the hippocampus that align with medical research, highlighting the power of this work for improving healthcare outcomes.