  We consider the $AdS_{3}/CFT_{2}$ duality between certain coset WZW theories
at large central charge and Vasiliev 3D higher spin gravity with a single
complex field. On the gravity side, we discuss a higher spin black hole
solution with chemical potential coupled to the spin-4 charge. We compute the
perturbative expansion of the higher spin charges and of the partition function
at high order in the chemical potential. The result is obtained with its exact
dependence on the parameter $\lambda$ characterising the symmetry algebra
$\mbox{hs}[\lambda]$. The cases of $\lambda=0,1$ are successfully compared with
a CFT calculation. The special point $\lambda=\infty$, the
Bergshoeff-Blencowe-Stelle limit, is also solved in terms of the exact
generating function for the partition function. The thermodynamics of both the
spin-4 and the usual spin-3 black holes is studied in order to discuss the
$\lambda$ dependence of the BTZ critical temperature $T_{\rm BTZ}(\lambda)$. In
the spin-3 case, it is shown that $T_{\rm BTZ}(\lambda)$ converges for large
$\lambda$ to the critical point of the $\lambda=\infty$ known partition
function previously found by the authors. In the spin-4 black hole, the picture
is qualitatively similar and $T_{\rm BTZ}(\infty)$ is accurately determined by
various numerical methods.
