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How can I change the format of the cell, which is currently being evaluated?

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One possibility is to use CellProlog/CellEpilog:

SetOptions[
    EvaluationNotebook[],
    StyleDefinitions -> Notebook[
        {
        Cell[StyleData[StyleDefinitions->"Default.nb"]],
        Cell[StyleData["Input"],
            CellProlog:>SetOptions[EvaluationCell[], Background->RGBColor[1,0,0]],
            CellEpilog:>SetOptions[EvaluationCell[], Background->None]
        ]
        },
        StyleDefinitions->"PrivateStylesheetFormatting.nb"
    ]
]
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