  While topological quantum computation is intrinsically fault-tolerant at zero
temperature, it loses its topological protection at any finite temperature. We
present a scheme to protect the information stored in a system supporting
non-cyclic anyons against thermal and measurement errors. The correction
procedure builds on the work of G\'acs [Gacs 1986] and Harrington [Harrington
2004] and operates as a local cellular automaton. In contrast to previously
studied schemes, our scheme is valid for both abelian and non-abelian anyons
and accounts for measurement errors. We analytically prove the existence of a
fault-tolerant threshold for a certain class of non-Abelian anyon models, and
numerically simulate the procedure for the specific example of Ising anyons.
The result of our simulations are consistent with a threshold between $10^{-4}$
and $10^{-3}$.
