Augmented Lagrangian method is an effective algorithm for solving convex optimization problems with linear constraints. While the primal subproblem needs to be solved iteratively, a common technique is to linearize the objective function of the subproblem and add a regularization term (equivalently to add a proximal term on the objective function of the subproblem), which is called the linearized augmented Lagrangian method. The proper selection of the regularization parameter is crucial to the convergence and convergence rate of the algorithm. The larger parameter can ensure the convergence of the algorithm, but it may lead to small stepsizes. The smaller parameter, however, permits larger stepsizes, but it may lead to divergence. In this paper, we consider the convex optimization problems with linear equality or inequality constraints. We design a class of adaptive indefinite linearized augmented Lagrangian method, that is, we use the information of the current iteration point to select the appropriate parameter of the regularization term adaptively, and try to make the primal stepsize larger with the guarantee of convergence, so as to improve the convergence rate of the algorithm. We theoretically prove the global convergence of the algorithm, and use numerical experiments to illustrate the effectiveness of the algorithm.