Let $E$, $X$ be irreducible smooth algebraic varieties over the complex numbers and let $p \colon E \to X$ be a morphism which is locally trivial with respect to the Zariski topology. Since $p$ is a vector bundle and the varieties are smooth, we get a natural isomorphism $p^\ast \colon\textrm{Pic}(X) \to \textrm{Pic}(E)$ by the pull-back of divisors, see e.g. Chp. 3 in [Fu84].

Is it true that $p^\ast \omega_X = \omega_E$ where $\omega_X$ and $\omega_E$ denote the canonical divisors of $X$ and $E$ respectively?

Any proof, counter-example or textbook reference would be perfect.

[Fu84] W. Fulton. Intersection theory, Springer-Verlag, Berlin, 1984.