Does GridSearchCV not support multi-class?

From the documentation on roc_auc_score:

Note: this implementation is restricted to the binary classification task or multilabel classification task in label indicator format.

By "label indicator format", they mean each label value is represented as a binary column (rather than as a unique target value in a single column). You don't want to do that for your predictor, as it could result in non-mutually-exclusive predictions (i.e., predicting both label 2 and 4 for case p1, or predicting no labels for case p2).

Pick or custom-implement a scoring function that is well-defined for the multiclass problem, such as F1 score. Personally I find informedness more convincing than F1 score, and easier to generalize to the multiclass problem than roc_auc_score.