Planar Gaussian fields are a model of spatial noise, and in many applications it is useful to understand the geometric structure of their level sets. There is a natural classification of geometric functionals of the level sets as either `local' (e.g. length of the level sets, volume of the excursion sets, Euler characteristic of the excursion sets) or 'non-local' (e.g. number of components of the level/excursion sets, percolation of the level/excursion sets) depending on whether there exists an integral representation for the functional. In the case of `local' functionals, first order properties (e.g. asymptotics for the mean) are easily derived from the Kac-Rice formula, and second order properties (e.g. asymptotics for the variance, central limit theorems) can also be established via Weiner chaos expansions (Kratz--Leon '11, Estrade--Leon '16, Marinucci--Rossi--Wigman '17, Nourdin--Peccati--Rossi '17 etc). For the `non-local' number of level/excursion sets the analysis is more challenging, and while first order properties were established 10 years ago by Nazarov--Sodin using ergodic theoretical techniques, up until now there have been essentially no second order results. In this talk I will discuss some first steps in this directions, namely proving lower bounds on the variance which are of `correct' order. Joint work with Dmitry Belyaev and Michael McAuley (University of Oxford).