  To realize the quantum anomalous Hall effect (QAHE) at elevated temperatures,
the approach of magnetic proximity effect (MPE) was adopted to break the
time-reversal symmetry in the topological insulator (Bi0.3Sb0.7)2Te3 (BST)
based heterostructures with a ferrimagnetic insulator europium iron garnet
(EuIG) of perpendicular magnetic anisotropy. Here we demonstrate phenomenally
large anomalous Hall resistance (RAHE) exceeding 8 {\Omega} (\r{ho}AHE of 3.2
{\mu}{\Omega}*cm) at 300 K and sustaining to 400 K in 35 BST/EuIG samples,
surpassing the past record of 0.28 {\Omega} (\r{ho}AHE of 0.14
{\mu}{\Omega}*cm) at 300 K. The remarkably large RAHE as attributed to an
atomically abrupt, Fe-rich interface between BST and EuIG. Importantly, the
gate dependence of the AHE loops shows no sign change with varying chemical
potential. This observation is supported by our first-principles calculations
via applying a gradient Zeeman field plus a contact potential on BST. Our
calculations further demonstrate that the AHE in this heterostructure is
attributed to the intrinsic Berry curvature. Furthermore, for gate-biased 4 nm
BST on EuIG, a pronounced topological Hall effect (THE) coexisting with AHE is
observed at the negative top-gate voltage up to 15 K. Interface tuning with
theoretical calculations has opened up new opportunities to realize
topologically distinct phenomena in tailored magnetic TI-based
heterostructures.
