Measurement of temperature is today popular and also in-expensive. You can have a temperature recorder based on a computer with date and time for about 100-200 Euro/$. Such equipment can also record humidity shocks etc. Recorder equipment can also be small with a diameter from only 2 cm.
For measuring food temperature and time have to be added together. And you can purchase TTI (=Time-Temperature-Indicators) from only 1 Euro/$.