  Currently, there is a great deal of interest in the seeking of consistent
thermodynamics of the Lorentzian Taub-NUT spacetimes. Despite a lot of
``satisfactory'' efforts have been made, all of these activities have been
restricted to the four-dimensional cases, with the higher even-dimensional
cases remaining unexplored. The aim of this article is to fill the gap, for the
first time. To the end of this subject, we first adopt our own idea that ``The
NUT charge is a thermodynamical multi-hair" to investigate the consistent
thermodynamics of $D = 6, 8, 10$ Lorentzian Taub-NUT spacetimes without a
cosmological constant. Similarly to the $D = 4$ cases as did in our previous
works, we find that the first law and Bekenstein-Smarr mass formulas are
perfectly satisfied if we still assign the secondary hair: $J_n = Mn$ as a
conserved charge in both mass formulae. Turning to the cases with a nonzero
cosmological constant, our treatment continues to work very well and all the
results can be fairly generalized to the corresponding Taub-NUT AdS spacetimes
in higher even-dimensions, although we do not know how to define and introduce
a similar higher-dimensional version of the dual (magnetic) mass that is well
known in four dimensions. Based upon the preceding results, we will also derive
the reduced version of the mass formulae when the secondary hair $J_n$ is
viewed as a redundant thermodynamic variable.
