  Minimal varying $\Lambda$ theories are defined by an action built from the
Einstein-Cartan-Holst first order action for gravity with the cosmological
constant $\Lambda$ as an independent scalar field, and supplemented by the
Euler and Pontryagin densities multiplied by $1/\Lambda$. We identify the
canonical structure of these theories which turn out to represent an example of
irregular systems. We find five degrees of freedom on generic backgrounds and
for generic values of parameters, whereas if the parameters satisfy a certain
condition (which includes the most commonly considered Euler case) only three
degrees of freedom remain. On de Sitter-like backgrounds the canonical
structure changes, and due to an emergent conformal symmetry one degree of
freedom drops from the spectrum. We also analyze the self-dual case with an
holomorphic action depending only on the self-dual part of the connection. In
this case we find two (complex) degrees of freedom, and further discuss the
Kodama state, the restriction to de Sitter background and the effect of reality
conditions.
