  We derive an exact formula for the boundary free energy of the open
Heisenberg XXZ spin chain. We allow for arbitrary boundary magnetic fields, but
assume zero bulk magnetization. The result is completely analogous to earlier
formulas for the so-called $g$-function: it is expressed as a combination of
single integrals and two simple Fredholm determinants. Our expressions can be
evaluated easily using numerical algorithms with arbitrary precision. We
demonstrate that the boundary free energy can show a wide variety of behaviour
as a function of the temperature, depending on the anisotropy and the boundary
fields. We also compute the low temperature limit of the boundary free energy,
and reproduce the known results for the ground state boundary energy, including
the case of non-diagonal fields.
