  The Feynman-Kac equation governs the distribution of the statistical
observable -- functional, having wide applications in almost all disciplines.
After overcoming challenges from the time-space coupled nonlocal operator and
the possible low regularity of functional, this paper develops the high-order
fully discrete scheme for the backward fractional Feynman-Kac equation by using
backward difference formulas (BDF) convolution quadrature in time, finite
element method in space, and some correction terms. With a systematic
correction, the high convergence order is achieved up to $6$ in time, without
deteriorating the optimal convergence in space and without the regularity
requirement on the solution. Finally, the extensive numerical experiments
validate the effectiveness of the high-order schemes.
