High-dimensional sparse modeling with censored survival data is of great practical importance, as exemplified by modern applications in high-throughput genomic data analysis and credit risk analysis. In this article, we propose a class of regularization methods for simultaneous variable selection and estimation in the additive hazards model, by combining the nonconcave penalized likelihood approach and the pseudoscore method. In a high-dimensional setting where the dimensionality can grow fast, polynomially or nonpolynomially, with the sample size, we establish the weak oracle property and oracle property under mild, interpretable conditions, thus providing strong performance guarantees for the proposed methodology. Moreover, we show that the regularity conditions required by the $L_1$ method are substantially relaxed by a certain class of sparsity-inducing concave penalties. As a result, concave penalties such as the smoothly clipped absolute deviation (SCAD), minimax concave penalty (MCP), and smooth integration of counting and absolute deviation (SICA) can significantly improve on the $L_1$ method and yield sparser models with better prediction performance. We present a coordinate descent algorithm for efficient implementation and rigorously investigate its convergence properties. The practical utility and effectiveness of the proposed methods are demonstrated by simulation studies and a real data example.