We consider a likelihood ratio method for testing whether a monotone baseline hazard function in the Cox model has a particular value at a fixed point. The characterization of the estimators involved is provided both in the nondecreasing and the nonincreasing setting. These characterizations facilitate the derivation of the asymptotic distribution of the likelihood ratio test, which is identical in the nondecreasing and in the nonincreasing case. The asymptotic distribution of the likelihood ratio test enables, via inversion, the construction of pointwise confidence intervals. Simulations show that these confidence intervals exhibit comparable coverage probabilities with the confidence intervals based on the asymptotic distribution of the nonparametric maximum likelihood estimator of a monotone baseline hazard function.