  We show that the statement in F. Sorge [Class. Quant. Grav. 36, no.23, 235006
(2019)] that the Casimir effect receives second order corrections due to
gravity is not consistent. We remark especially on the tracing of the proper
time Hamiltonian, where the correct procedure is to use the eigenfunctions and
eigenvalues of the covariant DAlembertian. After some cancellations we find
that the value of the functional W [0] is the same as obtained by Sorge.
However, we argue that the proper vacuum energy density carries extra
space-time volume terms that cancel over the gravitational correction,
returning to the same expression as in Minkowski space-time.
