  We derive the entropy product rule for
Taub-NUT~(Newman-Unti-Tamburino)-de~Sitter black hole~(BH) and
Taub-NUT--Anti-de~Sitter BH. We show that the entropy products in terms of both
the physical horizons are \emph{mass-independent}. Both \emph{perturbative}
approximation and \emph{direct} method have been considered. By introducing the
cosmological horizon we show that for Taub-NUT-de~Sitter BH, there exists a
mass-independent entropy functional relation in terms of three horizons namely
event horizon~(EH), Cauchy horizon~(CH) and cosmological horizon~(CHH) which
depends on cosmological parameter~($\Lambda$) and the NUT parameter~($N$). For
Taub-NUT-anti-de~Sitter BHs, we determine the mass-independent entropy
functional relations in terms of two physical horizons~(namely EH and CH) which
depends on only NUT parameter. Some-times some complicated functions of EH
entropy and CH entropy are also strictly mass-independent. This is plausible
only due to the new formalism developed in~\cite{wu}~[Phys. Rev. D 100,
101501(R)~(2019)] for NUT class of BHs. The formalism states that a generic
four-dimensional Taub-NUT spacetime should be described completely in terms of
three or four different types of thermodynamic hairs. They could be defined as
the Komar mass~($M=m$), the angular momentum~($J_{n}=mn$), the gravitomagnetic
charge ($N=n$), the dual~(magnetic) mass $(\tilde{M}=n)$. Finally, we could say
that this universality is mainly due to the presence of \emph{new conserved
charges $J_{N}=MN$} which is closely analogue to the Kerr-like angular momentum
$J=aM$.
