Barkhausen noise (BN) is a prototypical example of a complex system where the probability distributions for most relevant quantities follow a power-law behavior. In our experiment we investigated the role of temperature in determining the statistical properties of BN in thin Fe films. In the temperature range between 10 K and 295 K the probability distribution for the amplitude of the magnetization avalanches is always a power-law. The critical exponent a, however, undergoes a strong variation since its value changes from α = 1 at 295 K to α = 1.8 at 10 K. At our knowledge this is the first experimental evidence of the role of temperature in BN and, more generally, in complex systems. The experimental results are discussed in terms of a generalized version of the energy equipartition principle. Within this ansatz, the energy released during the dynamical evolution of the system is "shared" between all the available avalanches, depending on their size. Avalanches of a given size constitute a "mode" of evolution of the system: the energy globally released during the magnetization process is equally shared between all the available modes. In our experiment this behavior is actually observed at room temperature. At low temperature a "freezing" of the system prevents the occurrence of large size events and therefore energy is mostly released through jumps with small amplitude.