  It was recently reported that hydrogen-reduced PdCoO$_2$ films exhibit strong
perpendicular magnetic anisotropy (PMA) with sign tunable anomalous Hall effect
(AHE). Here, we provide extensive thickness-dependent study of this system, and
show that the electronic and magnetic properties are strongly dependent on the
thickness and annealing conditions. Below a critical thickness of 25 nm, AHE
shows clear PMA with hysteresis, and its sign changes from positive to
negative, and back to positive as the annealing temperature increases from 100
$^\circ$C to 400 $^\circ$C. Beyond the critical thickness, both PMA and AHE
hysteresis disappear and the AHE sign remains positive regardless of the
annealing parameters. Our results show that PMA may have a large role on AHE
sign-tunability and that below the critical thickness, competition between
different AHE mechanisms drives this sign change.
