  Inspired by the study of $V$-static manifold about classification, in this
article, we apply the recent results obtained by Freitas and Gomes (Compact
gradient Einstein-type manifolds with boundary, 2022) to prove the rigidity
results for compact gradient Einstein-type manifolds with nonempty boundary and
constant scalar curvature under some suitable pinching conditions. As a special
case of gradient Einstein-type manifold, we also give a rigidity result of
$(m,\rho)$-quasi-Einstein manifold with boundary.
