One thing was sure: he was convinced he had to save Anne's life at all costs.

at 在这里是介词，表示"以某种方式"或"处于某种状态"。 all 是形容词，表示"全部的、所有的"。 costs 是名词 cost 的复数形式，意思是"代价、成本"。 at all costs 是一个固定短语，字面意思是"以所有代价"，引申为"不惜一切代价、无论如何"。