  We calculate the eccentricity dependence of the high-order post-Newtonian
(PN) series for the generalized redshift invariant $\langle u^t \rangle_\tau$
for eccentric-orbit extreme-mass-ratio inspirals on a Schwarzschild background.
These results are calculated within first-order black hole perturbation theory
(BHPT) using Regge-Wheeler-Zerilli (RWZ) gauge. Our \textsc{Mathematica} code
is based on a familiar procedure, using PN expansion of the
Mano-Suzuki-Takasugi (MST) analytic function formalism for $l$ modes up to a
certain maximum and then using a direct general-$l$ PN expansion of the RWZ
equation for arbitrarily high $l$. We calculate dual expansions in PN order and
in powers of eccentricity, reaching 10PN relative order and $e^{20}$. Detailed
knowledge of the eccentricity expansion at each PN order allows us to find
within the eccentricity dependence numerous closed-form expressions and
multiple infinite series with known coefficients. We find leading logarithm
sequences in the PN expansion of the redshift invariant that reflect a similar
behavior in the PN expansion of the energy flux to infinity. A set of flux
terms and special functions that appear in the energy flux, like the
Peters-Mathews flux itself, are shown to reappear in the redshift PN expansion.
