  We investigate the thermal equation of state, bulk modulus, thermal expansion
coefficient, and heat capacity of MH-III (CH$_4$ filled-ice Ih), needed for the
study of CH$_4$ transport and outgassing for the case of Titan and
super-Titans. We employ density functional theory and ab initio molecular
dynamics simulations in the generalized-gradient approximation with a van der
Waals functional. We examine the finite temperature range of $300$K-$500$K and
pressures between $2$GPa-$7$GPa. We find that in this P-T range MH-III is less
dense than liquid water.
  There is uncertainty in the normalized moment of inertia (MOI) of Titan; it
is estimated to be in the range of $0.33-0.34$. If Titan's MOI is $0.34$,
MH-III is not stable at present in Titan's interior, yielding an easier path
for the outgassing of CH$_4$. However, for an MOI of $0.33$, MH-III is
thermodynamically stable at the bottom of a ice-rock internal layer capable of
storing CH$_4$. For rock mass fractions $\lessapprox 0.2$ upwelling melt is
likely hot enough to dissociate MH-III along its path. For super-Titans
considering a mixture of MH-III and ice VII, melt is always positively buoyant
if the H$_2$O:CH$_4$ mole fraction is $>5.5$. Our thermal evolution model shows
that MH-III may be present today in Titan's core, confined to a thin ($\approx
10$km) outer shell. We find that the heat capacity of MH-III is higher than
measured values for pure water-ice, larger than heat capacity often adopted for
ice-rock mixtures with implications for internal heating.
