The hardness and ideal strength of P-carbon, i.e., a new carbon phase for the cold-compressed carbon with an orthogonal structure recently proposed and named as P-carbon, are investigated by means of first-principles calculations. The strength calculations reveal that the failure mode in P-carbon is dominated by the tensile type. The ideal tensile strength of P-carbon is calculated to be 76.7GPa in the  direction, which is higher than that of the previously known most stable Z-carbon, of 71.4GPa. Meanwhile, the theoretical Vickers hardness of P-carbon is estimated as 89GPa, which is comparable with that of diamond. Especially, two types of bonds in P-carbon with hardness values of 114GPa and 105GPa are significantly stronger than those of diamond. The
results provide insight into exploration of the ultra-hard P-carbon for potentially technological applications.