  The topological Kondo effect arises when conduction electrons in metallic
leads are coupled to a mesoscopic superconducting island with Majorana
fermions. Working with its minimal setup, we study the lead electron local
tunneling density of states in its thermally smeared form motivated by scanning
tunneling microscopy, focusing on the component $\rho_{2 k_F}$ oscillating at
twice the Fermi wavenumber. As a function of temperature $T$ and at zero bias,
we find that the amplitude of $\rho_{2 k_F}$ is nonmonotonic, whereby with
decreasing $T$ an exponential thermal-length-controlled increase, potentially
through an intermediate Kondo logarithm, crosses over to a $T^{1/3}$ decay. The
Kondo logarithm is present only for tip-junction distances sufficiently smaller
than the Kondo length, thus providing information on the Kondo screening cloud.
The low temperature decay indicates non-Fermi-liquid scattering, in particular
the complete suppression of single-particle scattering at the topological Kondo
fixed point. For temperatures much below the Kondo temperature, we find that
the $\rho_{2 k_F}$ amplitude can be described as a universal scaling function
indicative of strong correlations. In a more general context, our
considerations point towards the utility of $\rho_{2 k_F}$ in studying quantum
impurity systems, including extracting information about the single-particle
scattering matrix.
