Under the assumption that the true density is decreasing, it is well known that the Grenander estimator converges at rate n1/3 if the true density is curved [Sankhy. Ser. A 31 (1969) 23-36] and at rate n1/2 if the density is flat [Ann. Probab. 11 (1983) 328-345; Canad. J. Statist. 27 (1999) 557-566]. In the case that the true density is misspecified, the results of Patilea [Ann. Statist. 29 (2001) 94-123] tell us that the global convergence rate is of order n1/3 in Hellinger distance. Here, we show that the local convergence rate is n1/2 at a point where the density is misspecified. This is not in contradiction with the results of Patilea [Ann. Statist. 29 (2001) 94-123]: the global convergence rate simply comes from locally curved well-specified regions. Furthermore, we study global convergence under misspecification by considering linear functionals. The rate of convergence is n1/2 and we show that the limit is made up of two independent terms: a mean-zero Gaussian term and a second term (with nonzero mean) which is present only if the density has well-specified locally flat regions.