  The first law of black hole mechanics, which relates the change of energy to
the change of entropy and other conserved charges, has been the main motivation
for probing the thermodynamic properties of black holes. In this work, we
investigate the thermodynamics of Kerr Taub-NUT AdS black holes. We present
geometric Komar definitions for the black hole charges, that by construction
satisfy the Smarr formula. Further, by a scaling argument based on Euler's
theorem, we establish the first law for the Kerr Taub-NUT AdS black holes. The
corresponding first law includes variations in the cosmological constant, NUT
charges and angular momenta. The key new ingredient in the construction are the
independent variations of both angular momenta, the black hole and Misner
string angular momenta. Employing the Brown-York quasilocal charge definitions
we show that our expression for the mass and spin coincide with our generalized
Komar expressions. We indicate the relevance of these results to the
thermodynamics of rotating AdS black holes, including the proper choice of
time-like Killing vector to produce the correct thermodynamic mass.
