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Temperature is a key determinant of microbial behaviour and survival in the environment and within hosts. At intermediate temperatures, growth rate varies according to the Arrhenius law of thermodynamics, which describes the effect of temperature on the rate of a chemical reaction. However, the mechanistic basis for this behaviour remains unclear. Here we use single-cell microscopy to show that Escherichia coli exhibits a gradual response to temperature upshifts with a timescale of ~1.5 doublings at the higher temperature. The response was largely independent of initial or final temperature and nutrient source. Proteomic and genomic approaches demonstrated that adaptation to temperature is independent of transcriptional, translational or membrane fluidity changes. Instead, an autocatalytic enzyme network model incorporating temperature-sensitive Michaelis–Menten kinetics recapitulates all temperature-shift dynamics through metabolome rearrangement, resulting in a transient temperature memory. The model successfully predicts alterations in the temperature response across nutrient conditions, diverse E. coli strains from hosts with different body temperatures, soil-dwelling Bacillus subtilis and fission yeast. In sum, our model provides a mechanistic framework for Arrhenius-dependent growth.