One of the key issues of modern days statistical physics is the investigation of complex systems, with a particular emphasis on the occurrence of power-law probability distributions for certain observable quantities x: P(x) = x^(-a). A major goal of these investigations is the possibility to predict, on the basis of the microscopic properties of the system, the value of the so-called critical exponent a. Surprisingly enough, none of the available paradigms explicitly takes into account one of the central concepts of classical statistical physics, i.e., temperature. In our experiment we studied the role of temperature in determining the statistical properties of Barkhausen noise in thin Fe films. In the temperature range between 10 K and 300 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 a = 1 at 300 K to a = 1.8 at 10 K. At our knowledge this is the first experimental evidence of the role of temperature in a complex system. In the present paper we suggest the possibility that for complex systems such as magnetic materials, a generalized version of the energy equipartition principle might hold. Within this ansatz the energy released during the dynamical evolution of the system is shared between all the available avalanches depending on their size: the amount of energy released through avalanches of each possible size is the same. In our experiment this behaviour is actually observed at room temperature. At low temperature a freezing of the system prevents the occurrence of large size events and therefore the critical exponent has a larger value. A similar effects, due to quantum effects, also takes place in classical statistical physics: the well known freezing of the rotational and vibrational degrees of freedom of a gas molucule at low temperature.