In a previous joint work with Berestycki, we established a Weyl law for the eigenvalue counting functions of subcritical Liouville quantum gravity (LQG) surfaces with leading coefficient proportional to the total LQG area. It is therefore very natural to ask whether a Weyl law also holds in the critical case. To address this problem, we must first examine the local behaviour of critical Gaussian multiplicative chaos (GMC). I will introduce power variations of critical GMC and discuss their asymptotic behaviour along refining partitions as well as their connections to supercritical chaos. We will see how this enables one to establish an optimal modulus of continuity for critical GMC, which in turn shows that zero lies in the essential spectrum of the critical LQG Laplacian, thereby answering the spectral question in the negative.