Inspired by the recent paper of Baltazar and Queiroz (Rigidity Theorem for Integral Pinched Static Manifolds and Related Critical Spaces, J. Geom. Anal. 2024), in this article, we prove the rigidity for compact gradient Einstein-type manifolds with nonempty boundary and constant scalar curvature under a pinching condition, which is independent on the potential function. As a special case of gradient Einstein-type manifold, we also give a rigidity result of (m, ρ)-quasi-Einstein manifold with boundary.
2020 Mathematics Subject Classification. 53C25; 53C20; 53C21.