Possible high-temperature superconductivity in multilayer graphane: can the cuprates be beaten?

We analyze a possible superconductivity in the hole-doped system of layered hydrogenized graphene by taking into account thermal fluctuations of the order parameter. In particular, we demonstrate that in the one-layer case the values of the high mean-field (MF) critical temperature $T_c^{MF}\sim 80-90K$, predicted recently by Savini et al \cite{Savini}, do not alter significantly due to the fluctuations, and the Berezinskii-Kosterlitz-Thouless (BKT) critical temperature of the vortex superconductivity is almost the same as the MF temperature at doping 0.01-0.1. We show that in the case of multilayer system, when the coupling between the layers stabilizes the superconducting phase in the form of fluxon superconductivity, the critical temperature $T_c$ can increase dramatically to the values $\sim 150K$, higher than the corresponding values in cuprates under ambient pressure.