  Phonon-phonon anharmonic effects have a strong influence on the phonon
spectrum; most prominent manifestation of these effects are the softening
(shift in frequency) and broadening (change in FWHM) of the phonon modes at
finite temperature. Using Raman spectroscopy, we studied the temperature
dependence of the FWHM and Raman shift of $\mathrm{E_{2g}^1}$ and
$\mathrm{A_{1g}}$ modes for single-layer and natural bilayer MoS$_2$ over a
broad range of temperatures ($8 < $T$ < 300$ K). Both the Raman shift and FWHM
of these modes show linear temperature dependence for $T>100$ K, whereas they
become independent of temperature for $T<100$ K. Using first-principles
calculations, we show that three-phonon anharmonic effects intrinsic to the
material can account for the observed temperature-dependence of the line-width
of both the modes. It also plays an important role in determining the
temperature-dependence of the frequency of the Raman modes. The observed
evolution of the line-width of the A$_{1g}$ mode suggests that electron-phonon
processes are additionally involved. From the analysis of the
temperature-dependent Raman spectra of MoS$_2$ on two different substrates --
SiO$_2$ and hexagonal boron nitride, we disentangle the contributions of
external stress and internal impurities to these phonon-related processes. We
find that the renormalization of the phonon mode frequencies on different
substrates is governed by strain and intrinsic doping. Our work establishes the
role of intrinsic phonon anharmonic effects in deciding the Raman shift in
MoS$_2$ irrespective of substrate and layer number.
