In this paper, we analyzed a priori and a posteriori error estimates for the h − p version of the finite element discretization of the elliptic Robin boundary control problem. The conforming $h−p$ finite element method is used. First, we established the optimality conditions for the continuous and discrete optimal control problems, respectively. Then, a priori error estimates of the $h−p$ finite element discretization for the optimal control problem are derived rigorously. Moreover, residual-based a posteriori error estimates are established for the coupled state and control approximations. Such estimators can be used to construct reliable adaptive methods for optimal control problems.