  We show that for general spherically symmetric configurations, contributions
of general gravitational and mixed gauge-gravitational Chern-Simons terms to
the equations of motion vanish identically in $D>3$ dimensions. This implies
that such terms in the action do not affect Birkhoff's theorem or any
previously known spherically symmetric solutions. Furthermore, we investigate
the thermodynamical properties using the procedure described in an accompanying
paper. We find that in $D>3$ static spherically symmetric case Chern-Simons
terms do not contribute to the entropy either. Moreover, if one requires only
for the metric tensor to be spherically symmetric, letting other fields
unrestricted, the results extend almost completely, with only one possible
exception --- Chern-Simons Lagrangian terms in which the gravitational part is
just the $n=2$ irreducible gravitational Chern-Simons term.
