Why is the light emitted by the LED not of a single wavelength but consists of a narrow range of wavelength instead? Spectral linewidth change?
Each photon emitted from an LED is the product of an electron-hole pair recombination. Charge carriers (electrons/holes) in a semiconductor have an energy distribution which is a function of dopant concentration, density of states of the semiconductor and the Fermi distribution. At elevated temperature the Fermi distribution is stretched out and results in a wider range of electron and hole energies. These are all still concentrated near the bandgap, but increases the probability of a slightly more energetic electron combining with a slightly more energetic hole to emit a photon of higher energy (shorter wavelength). At higher temperatures there are also more phonons available that enable transitions that would otherwise be forbidden. This also broadens the emitted spectrum and allow for slightly sub-bandgap emission.
But temperature also affects the bandgap energy. As the temperature increases, the bandgap energy decreases. This will shift your peak to longer wavelengths at higher temperature.