I will present a third order accurate finite volume scheme under Adaptive Mesh Refinement (AMR) on quad-tree type grids. In the scheme, AMR is driven by the, so called, numerical entropy production. This is a residual of the entropy inequality that is computable in each space-time finite volume during the simulation and that is of the same size of the truncation error, which has been successfully exploited to control adaptive behavior of schemes in various ways in one and two space dimensions. Of course, the reconstruction of point values from cell averages requires a procedure that is third order accurate, non-oscillatory, but also versatile enough to handle data on unstructured and non-conforming grids and efficient in computing point data at very many points in each cell: the Central WENO (CWENO) technique is employed here for this task.