  We study whether the relations between the Weyl anomaly, entanglement entropy
(EE), and thermal entropy of a two-dimensional (2D) conformal field theory
(CFT) extend to 2D boundaries of 3D CFTs, or 2D defects of $D \geq 3$ CFTs. The
Weyl anomaly of a 2D boundary or defect defines two or three central charges,
respectively. One of these, $b$, obeys a c-theorem, as in 2D CFT. For a 2D
defect, we show that another, $d_2$, interpreted as the defect's `conformal
dimension,' must be non-negative by the Averaged Null Energy Condition (ANEC).
We show that the EE of a sphere centered on a planar defect has a logarithmic
contribution from the defect fixed by $b$ and $d_2$. Using this and known
holographic results, we compute $b$ and $d_2$ for 1/2-BPS surface operators in
the maximally supersymmetric (SUSY) 4D and 6D CFTs. The results are consistent
with $b$'s c-theorem. Via free field and holographic examples we show that no
universal `Cardy formula' relates the central charges to thermal entropy.
