Nanoparticles are widely used in composite materials, nanoscale devices, biological detectors and medical treatment. The thermophysical properties of a single nanoparticle are, therefore, important for both nanotechnology and nanoscience applications. However, property measurements are limited by the spatial resolution of conventional measurement methods, so there are not yet any effective measurement methods to characterize the thermophysical properties of a single nanoparticle. This paper describes a laser flash Raman spectroscopy method for measuring the specific heat of a single nanoparticle supported on a free-standing substrate based on a lumped parameter model for the nanoparticle coupled with a transient 2D thermal conduction model for the suspended substrate. A series of square laser pulses are assumed to be used to heat the supported single nanoparticle in a vacuum. The temperature increases in the single nanoparticle and the suspended substrate are then measured based on their Raman band shifts. The laser absorption coefficients of the nanoparticle and the substrate are then eliminated by comparing the temperature increases measured using different laser pulse widths. The specific heat of the nanoparticle and the thermal contact conductance between the nanoparticle and the substrate can then be extracted by fitting the temperatures of both the nanoparticle and the substrate. Case studies show that the method can accurately measure the specific heat of a single nanoparticle about 100 nm in diameter using ~1 ns pulse widths. The influence of the nanoparticle geometry and the thermophysical properties of the substrate are also discussed.