  We study the thermodynamic properties of modified spin-$S$ Kitaev models
introduced by Baskaran, Sen and Shankar (Phys. Rev. B 78, 115116 (2008)). These
models have the property that for half-odd-integer spins their eigenstates map
on to those of spin-1/2 Kitaev models, with well-known highly entangled quantum
spin-liquid states and Majorana fermions. For integer spins, the Hamiltonian is
made out of commuting local operators. Thus, the eigenstates can be chosen to
be completely unentangled between different sites, though with a significant
degeneracy for each eigenstate. For half-odd-integer spins, the thermodynamic
properties can be related to the spin-1/2 Kitaev models apart from an
additional degeneracy. Hence we focus here on the case of integer spins. We use
transfer matrix methods, high temperature expansions and Monte Carlo
simulations to study the thermodynamic properties of ferromagnetic and
antiferromagnetic models with spin $S=1$ and $S=2$. Apart from large residual
entropies, which all the models have, we find that they can have a variety of
different behaviors. Transfer matrix calculations show that for the different
models, the correlation lengths can be finite as $T\to 0$, become critical as
$T\to 0$ or diverge exponentially as $T\to 0$. There is a conserved $Z_2$ flux
variable associated with each hexagonal plaquette which saturates at the value
$+1$ as $T\rightarrow0$ in all models except the $S=1$ antiferromagnet where
the mean flux remains zero as $T\to 0$. We provide qualitative explanations for
these results.
