Both DifferentialRoot
and DifferenceRoot
use the same template for their box structure, which is named "HolonomicDisplay".

Therefore, the styling is set by the stylesheet, which you can access with CurrentValue[{StyleDefinitions, "HolonomicDisplay"}]
, and also change in the usual manner of changing stylesheet. You can also set it programatically:
displayFunction = (TemplateBoxOptionsDisplayFunction /.
CurrentValue[{StyleDefinitions,
"HolonomicDisplay"}]) /. (Background -> _) -> (Background -> Yellow);
SetOptions[SelectedNotebook[],
StyleDefinitions ->
Notebook[{Cell[StyleData[StyleDefinitions -> "Default.nb"]],
Cell[StyleData["HolonomicDisplay"],
TemplateBoxOptions -> {InterpretationFunction -> (#2 &),
DisplayFunction -> displayFunction}]}]];

Version 13.1+
It seems that TemplateBoxOptionsDisplayFunction
was changed to TemplateBoxOptions
in 13.1. Therefore, you should instead use:
displayFunction = (TemplateBoxOptions /.
CurrentValue[{StyleDefinitions,
"HolonomicDisplay"}]) /. (Background -> _) -> (Background -> Yellow);