We revisit the study of the phase structure of higher spin black holes carried out in arXiv$:1210.0284$ using the canonical formalism. In particular we study the low as well as high temperature regimes. We show that the Hawking-Page transition takes
place in the low temperature regime. The thermodynamically favoured phase changes from conical surplus to black holes and then again to conical surplus as we increase temperature. We then show that in the high temperature regime the diagonal embedding gives the appropriate description. We also give a map between the parameters of the theory near the IR and UV fixed points. This makes the good solutions near one end map to the bad solutions near the other end and vice versa.
It has recently been argued that, classically, massless higher spin theories in AdS_3 have an enlarged W_N-symmetry as the algebra of asymptotic isometries. In this note we provide evidence that this symmetry is realised (perturbatively) in the quant
um theory. We perform a one loop computation of the fluctuations for a massless spin $s$ field around a thermal AdS_3 background. The resulting determinants are evaluated using the heat kernel techniques of arXiv:0911.5085. The answer factorises holomorphically, and the contributions from the various spin $s$ fields organise themselves into vacuum characters of the W_N symmetry. For the case of the hs(1,1) theory consisting of an infinite tower of massless higher spin particles, the resulting answer can be simply expressed in terms of (two copies of) the MacMahon function.
