  We calculate the high-order post-Newtonian (PN) expansion of the energy and
angular momentum fluxes onto the horizon of a nonspinning black hole primary in
eccentric-orbit extreme-mass-ratio inspirals. The first-order black hole
perturbation theory calculation uses \textsc{Mathematica} and makes an analytic
expansion of the Regge-Wheeler-Zerilli functions using the Mano-Suzuki-Takasugi
formalism. The horizon absorption, or tidal heating and torquing, is calculated
to 18PN relative to the leading horizon flux (i.e., 22PN order relative to the
leading quadrupole flux at infinity). Each PN term is a function of
eccentricity $e$ and is calculated as a series to $e^{10}$. A second expansion,
to 10PN horizon-relative order (or 14PN relative to the flux at infinity), is
computed deeper in eccentricity to $e^{20}$. A number of resummed closed-form
functions are found for the low PN terms in the series. Using a separate
Teukolsky perturbation code, numerical comparisons are made to test how
accurate the PN expansion is when extended to a close $p=10$ orbit. We find
that the horizon absorption expansion is not as convergent as a previously
computed infinity-side flux expansion. However, given that the horizon
absorption is suppressed by 4PN, useful results can be obtained even with an
orbit as tight as this for $e \le 1/2$. Combining the present results with our
earlier expansion of the fluxes to infinity makes the knowledge of the total
dissipation known to 19PN for eccentric-orbit nonspinning EMRIs.
