  The first law of black hole thermodynamics in the presence of a cosmological
constant $\Lambda$ can be generalized by introducing a term containing the
variation $\delta \Lambda$. Similar to other terms in the first law, which are
variations of some conserved charges like mass, entropy, angular momentum,
electric charge etc., it has been shown in [1] that the new term has the same
structure: $\Lambda$ is a conserved charge associated with a gauge symmetry;
and its role in the first law is quite similar to an "electric charge" rather
than to the pressure. Besides, its conjugate chemical potential resembles an
"electric potential" on the horizon, in contrast with the volume enclosed by
horizon. In this work, first we propose and prove the generalized Smarr
relation in this new paradigm. Then, we reproduce systematically the "effective
volume" of a black hole which has been introduced before in the literature as
the conjugate of pressure. Our construction removes the ambiguity in the
definition of volume. Finally, we apply and investigate this formulation of
"$\Lambda$ as a charge" on a number of solutions to different models of gravity
for different spacetime dimensions. Specially, we investigate the applicability
and validity of the analysis for black branes, whose enclosed volume is not
well-defined in principle.
