  The Barbero-Immirzi parameter $\gamma$ appears in the \emph{real} connection
formulation of gravity in terms of the Ashtekar variables, and gives rise to a
one-parameter quantization ambiguity in Loop Quantum Gravity. In this paper we
investigate the conditions under which $\gamma$ will have physical effects in
Euclidean Quantum Gravity. This is done by constructing a well-defined
Euclidean path integral for the Holst action with non-zero cosmological
constant on a manifold with boundary. We find that two general conditions must
be satisfied by the spacetime manifold in order for the Holst action and its
surface integral to be non-zero: (i) the metric has to be non-diagonalizable;
(ii) the Pontryagin number of the manifold has to be non-zero. The latter is a
strong topological condition, and rules out many of the known solutions to the
Einstein field equations. This result leads us to evaluate the on-shell
first-order Holst action and corresponding Euclidean partition function on the
Taub-NUT-ADS solution. We find that $\gamma$ shows up as a finite rotation of
the on-shell partition function which corresponds to shifts in the energy and
entropy of the NUT charge. In an appendix we also evaluate the Holst action on
the Taub-NUT and Taub-bolt solutions in flat spacetime and find that in that
case as well $\gamma$ shows up in the energy and entropy of the NUT and bolt
charges. We also present an example whereby the Euler characteristic of the
manifold has a non-trivial effect on black-hole mergers.
