  We calculate the bulk and boundary parts of the free energy for an open
spin-1/2 XXZ-chain in the critical regime by bosonisation. We identify the
cutoff independent contributions and determine their amplitudes by comparing
with Bethe ansatz calculations at zero temperature T. For the bulk part of the
free energy we find agreement with Lukyanov's result [Nucl.Phys.B 522, 533
(1998)]. In the boundary part we obtain a cutoff independent term which is
linear in T and determines the temperature dependence of the boundary
susceptibility in the attractive regime for $T\ll 1$. We further show that at
particular anisotropies where contributions from irrelevant operators with
different scaling dimensions cross, logarithmic corrections appear. We give
explicit formulas for these terms at those anisotropies where they are most
important. We verify our results by comparing with extensive numerical
calculations based on a numerical solution of the T=0 Bethe ansatz equations,
the finite temperature Bethe ansatz equations in the quantum-transfer matrix
formalism, and the density-matrix renormalisation group applied to transfer
matrices.
