  We study the method of extended gravitational decoupling in obtaining static
black hole solutions satisfying Einstein's equations with a tensor vacuum. The
source has quite generic characteristics and satisfies the strong energy
condition. The stationary, axisymmetric counterpart of the static metric is
obtained by applying the Newman-Janis and Azreg-A\"{i}nou algorithms. The
thermodynamics of the rotating solution is studied and the expressions of
various thermodynamic quantities are derived. The dependence of the
temperature, free energy and specific heat on the horizon radius is studied for
various values of the hairy parameter and the black hole spin. Such a study
reveals that small hairy black holes are thermodynamically more stable compared
to large hairy black holes, and that the horizon radius and temperature range
for which the rotating hairy black holes can be in thermodynamic equilibrium
with the surroundings depends non-trivially on the hairy parameters. We further
discuss the first law of black hole thermodynamics for the hairy case and
discuss its implications.
