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Within the framework of finite temperature DFT, we present and assess a novel multiscale recursion method to calculate the electronic density and the local density of states via the Green's function. Unlike plane wave methods, the computational time decreases with temperature. Tests are performed on a system representing a hydrogen plasma with a local pseudopotential. Calculations are distributed on grids with different spacings through scaling properties of the recursion. The computational workload increases linearly with the size of the system. It can be productively dispatched on an arbitrary number of processors. This method opens an angle to molecular dynamics simulations of hydrogen plasma in a range of material density and temperature where plane wave methods fail to compute efficiently the electronic density.