We present a new DFTB-p3b density functional tight binding model for hydrogen at extremely high pressures and temperatures, which includes a polarizable basis set (p) and a three-body environmentally dependent repulsive potential (3b). We find that use of an extended basis set is necessary under dissociated liquid conditions to account for the substantial p-orbital character of the electronic states around the Fermi energy. The repulsive energy is determined through comparison to cold curve pressures computed form Density Functional Theory (DFT) for the hexagonal close packed solid, as well as pressures from thermally equilibrated DFT-MD simulations of the liquid phase. In particular, we observe improved agreement in our DFTB-p3b model with previous theoretical and experimental results for the shock Hugoniot of hydrogen up to 100 GPa and 25,000 K, compared to a standard DFTB model using pair-wise interactions and an s-orbital basis set, only. The DFTB-p3b approach discussed here provides a general method to extend the DFTB method for a wide variety of materials over a significantly larger range of thermodynamic conditions than previously possible.