A cold storage heat exchanger has refrigerant pipes and a cold storage material. Each of the refrigerant pipes has a refrigerant passage therein. The refrigerant pipes are arranged so as to be distanced from each other with clearances therebetween. The cold storage material is disposed in at least one of the clearances. At least another one of the clearances provides an air passage. The heat storage material is disposed in at least two of the clearances arranged adjacent to each other.