Gold nanoparticles emit broad-band upconverted luminescence upon irradiation with pulsed infrared laser radiation. Although the phenomenon is widely observed, considerable disagreement still exists concerning the underlying physics, most notably over the applicability of concepts such as multiphoton absorption, inelastic scattering, and interband vs intraband electronic transitions. Here, we study single particles and small clusters of particles by employing a spectrally resolved power-law analysis of the irradiation-dependent emission as a sensitive probe of these physical models. Two regimes of emission are identified. At low irradiance levels of kW/cm, the emission follows a well-defined integer-exponent power law suggestive of a multiphoton process. However, at higher irradiance levels of several kW/cm, the nonlinearity exponent itself depends on the photon energy detected, a tell-tale signature of a radiating heated electron gas. We show that in this regime, the experiments are incompatible with both interband transitions and inelastic light scattering as the cause of the luminescence, whereas they are compatible with the notion of luminescence linked to intraband transitions.