A composite oxide with high diffusion rate of lithium is provided. Alternatively, a lithium-containing complex phosphate with high diffusion rate of lithium is provided. Alternatively, a positive electrode active material with high diffusion rate of lithium is provided. Alternatively, a lithium ion battery with high output is provided. Alternatively, a lithium ion battery that can be manufactured at low cost is provided. A positive electrode active material is formed through a first step of mixing a lithium compound, a phosphorus compound, and water, a second step of adjusting pH by adding a first aqueous solution to a first mixed solution formed in the first step, a third step of mixing an iron compound with a second mixed solution formed in the second step, a fourth step of performing heat treatment under a pressure more than or equal to 0.1 MPa and less than or equal to 2 MPa at a highest temperature more than 100° C. and less than or equal to 119° C. on a third mixed solution formed in the third step with a pH of more than or equal to 3.5 and less than or equal to 5.0.