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We compute exact second-order asymptotics for the cost of an optimal solution to the entropic optimal transport problem in the continuous-to-discrete, or semidiscrete, setting. In contrast to the discrete-discrete or continuous-continuous case, we show that the first-order term in this expansion vanishes but the second-order term does not, so that in the semidiscrete setting the difference in cost between the unregularized and the regularized solutions is quadratic in the inverse regularization parameter, with a leading constant that depends explicitly on the value of the density at the points of discontinuity of the optimal unregularized map between the measures. We develop these results by proving new pointwise convergence rates of the solutions to the dual problem, which may be of independent interest. [ABSTRACT FROM AUTHOR]