Running variable-resolution climate simulations with regional refinement is an important application. Recently, the nonhydrostatic dynamical core of the Model for Prediction Across Scales (MPAS), which supports unstructured, variable resolution grids, has been implemented within the Community Earth System Model (CESM). However, it is known that there is a dependence of horizontal resolution and physics time step on model results, since the Zhang-McFarlane (ZM) deep convection scheme in the Community Atmosphere Model (CAM) physics is constrained by a convective relaxation time scale which does not remove instabilities in a short time step. This raises a question of whether a spatially and temporally constant value of the time scale is appropriate for higher resolution simulations, particularly for variable-resolution modeling in which the grid spacing is not uniform globally. To address this issue, we implemented the Grell-Freitas (GF) scale-aware stochastic deep convection scheme into the CAM physics. The GF scheme uses multiple formulations of the quasi-equilibrium closures, and the relaxation time-scale is a function of the mesh size and vertical velocity scale. The effects of the two deep convection schemes on the simulated characteristics of precipitation will be compared in aquaplanet simulations with quasi-uniform (120 km) and variable-resolution (120-30 km) meshes to document and understand differences in the model behaviors. It was found that the GF scheme is less sensitive to the physics time step than the ZM scheme. In addition, deep convection is more active in the GF simulations, and the GF scheme is less sensitive to grid size than the ZM scheme.