  We study subleading corrections to the corner free energy in classical
two-dimensional critical systems, focusing on a generic boundary perturbation
by the stress-tensor of the underlying conformal field theory (CFT). In the
particular case of an angle $2\pi$, we find that there is an unusual correction
of the form $L^{-1}\log L$, where $L$ is a typical length scale in the system.
This correction also affects the one-point function of an operator near the
corner. The prefactor can be seen as semi-universal, in the sense that it
depends on a \emph{single} non-universal quantity, the extrapolation length.
Once this ultraviolet cutoff is known, the term is entirely fixed by the
geometry of the system, and the central charge of the CFT. Such a corner
appears for example in the bipartite fidelity of a one-dimensional quantum
system at criticality, which allows for several numerical checks in free
fermions systems. We also present an exact result in the XX and Ising chains
that confirms this analysis. Finally, we consider applications to the time
evolution of the (logarithmic) Loschmidt echo and the entanglement entropy
following a local quantum quench. Due to subtle issues in analytic
continuation, we find that the logarithmic term in imaginary time transforms
into a time-dependent $L^{-2}$ correction for the entanglement entropy, and a
$L^{-2}\log L$ term for the Loschmidt echo.
