Improving the description of the equation of state (EoS) of deuterium-tritium (DT) has recently been shown to change significantly the gain of an Inertial Confinement Fusion (ICF) target (Hu et al., PRL 104, 235003 (2010)). We use here an advanced multi-phase equation of state (EoS), based on ab initio calculations, to perform a full optimization of the laser pulse shape with hydrodynamic simulations starting from 19 K in DT ice. The thermonuclear gain is shown to be a robust estimate over possible uncertainties of the EoS. Two different target designs are discussed, for shock ignition and self-ignition. In the first case, the areal density and thermonuclear energy can be recovered by slightly increasing the laser energy. In the second case, a lower in-flight adiabat is needed, leading to a significant delay (3ns) in the shock timing of the implosion.PACS numbers: 52.57. Bc, 47.40.Nm, 47.55.Kf, 52.57.Fg The DT EoS, starting from the cryogenic solid and undergoing a wide range of plasma conditions, is a key input to simulate the ICF implosion of the DT pellet and hence to quantify the fusion energy production. The current uncertainty in the EoS of DT, more particularly in the strongly correlated and degenerate regime, is maintaining uncertainties in hydro-simulations for the prediction of ICF thermonuclear gains and also for the shock timing design. Recently, Hu et al. [1] have shown a significant reduction (30 %) of the thermonuclear gain by improving the DT EoS in the strongly coupled and degenerate regime. A direct implication of this work is the reduction of the safety margin in current ICF designs that could be deleterious to achieve high gains. But, no re-optimization of the laser pulse has been performed with this partial ab initio EoS, only valid for T ≥ 1.35 eV. Consequently, that poses two important questions: can the decrease in thermonuclear energy be avoided by re-shaping the laser pulse, and what result would this have on the tuning of shock timing? The aim of the present work is to address these two questions.We use here a recently published Multi-Phase DT EoS[2], dubbed MP-EoS, that is based on ab initio calculations in the strongly coupled and degenerate regime (ρ = 0.5 − 12.5 g/cc and T ≤ 10 eV), and extended here to cover the whole thermodynamical path of DT ice (ρ = 0.25 -4.10 3 g/cc and T=0-10 keV). Hydrosimulations can thus start at the realistic temperature of 19 K for DT ice, that allows to take into account the effect of the solid-liquid transition and to avoid any effect of pre-heating of the shell. Two different target designs are considered, one for shock ignition and the other for self-ignition, corresponding to two different thermodynamic paths for the compression of the DT fuel. It will * Corresponding author: benoit.canaud@cea.fr be shown below that the energy gain of the target is in fact a robust estimate, that can be recovered after optimization of the laser pulse. But the use of a more realistic EoS can lead to large changes in the shock timing.In classical Inertial Confine...