It's not worth getting into an argument with Tim, as I learned to my cost.

cost 在这里是名词，本义是"成本、费用"，引申为"代价、损失"。 learn to one's cost 是一个固定短语，字面意思是"通过付出代价而学到"，实际表示"吃了苦头才明白"的意思。