Let's say we have a vector bundle $V$ on a compact Riemannian manifold $M$ (of dimension $m$, with metric $g$). Given a differential operator $P$, acting on the sections of $V$, of the form: $$P = \nabla^* \nabla + E,$$ where $\nabla$ is a connection on $V$, with $\nabla^* \nabla$ being the connection Laplacian, and $E$ is an endomorphism of $V$.

Is it necessary for $E$ to be invariant under a diffeomorphism $\phi : M \rightarrow M$?

In particular, under general covariance (invariance of the form under arbitrary differentiable coordinate transformations)?

For more context see:

1. Section 11.2 of http://www.alainconnes.org/docs/bookwebfinal.pdf
2. Lemma 4.8.1 of https://pages.uoregon.edu/gilkey/dirPDF/InvarianceTheory1Ed.pdf