Another possibility is storing the data in a serial EEPROM. Two temperatures will be 4 bytes. The date will be 3 or 4 bytes (and there are ways of storing the data that would use a lot less). So, 8 bytes every 5 minutes is 71kB a month. You can store more than 3 months of data in a 2Mbit (=256kbyte) SPI EEPROM and add a serial real-time clock such as the DS3234.
Pete