Abstract A regularization-independent universal formula for the energy–momentum tensor in gauge theory in the flat spacetime can be written down by employing the so-called Yang–Mills gradient flow. We examine a possible use of the formula in the calculation of the axial anomaly in a gravitational field, the anomaly first obtained by Toshiei Kimura [Prog. Theor. Phys. 42, 1191 (1969)]. As a general argument indicates, the formula reproduces the correct non-local structure of the (axial current)–(energy–momentum tensor)–(energy–momentum tensor) triangle diagram in a way that is consistent with the axial anomaly. On the other hand, the formula does not automatically reproduce the general coordinate (or translation) Ward–Takahashi relation, requiring corrections by local counterterms. This analysis thus illustrates the fact that the universal formula as it stands can be used only in on-shell correlation functions, in which the energy–momentum tensor does not coincide with other composite operators in coordinate space.