  We explore the decoherence of the gapless/critical boundary of a topological
order, through interactions with the bulk reservoir of "ancilla anyons." We
take the critical boundary of the $2d$ toric code as an example. The intrinsic
nonlocal nature of the anyons demands the strong and weak symmetry condition
for the ordinary decoherence problem be extended to the strong or weak gauge
invariance conditions. We demonstrate that in the $\textit{doubled}$ Hilbert
space, the partition function of the boundary is mapped to two layers of the
$2d$ critical Ising model with an inter-layer line defect that depends on the
species of the anyons causing the decoherence. The line defects associated with
the tunneling of bosonic $e$ and $m$ anyons are relevant, and result in
long-range correlations for either the $e$ or $m$ anyon respectively on the
boundary in the doubled Hilbert space. In contrast, the defect of the $f$ anyon
is marginal and leads to a line of fixed points with varying effective central
charges, and power-law correlations having continuously varying scaling
dimensions. We also demonstrate that decoherence-analogues of Majorana zero
modes are localized at the spatial interface of the relevant $e$ and $m$ anyon
decoherence channels, which leads to a universal logarithmic scaling of the
R\'enyi entropy of the boundary.
