  It is shown that the Dirac equation is separable by variables in a
five-dimensional rotating Kerr-(anti-)de Sitter black hole with two independent
angular momenta. A first order symmetry operator that commutes with the Dirac
operator is constructed in terms of a rank-three Killing-Yano tensor whose
square is a second order symmetric Stackel-Killing tensor admitted by the
five-dimensional Kerr-(anti-)de Sitter spacetime. We highlight the construction
procedure of such a symmetry operator. In addition, the first law of black hole
thermodynamics has been extended to the case that the cosmological constant can
be viewed as a thermodynamical variable.
