The celebrated Grenander (1956) estimator is the maximum likelihood estimator of a decreasing density function. In contrast to alternative nonparametric density estimators, Grenander estimator does not require any smoothing parameters and is often viewed as a fully automatic procedure. However, the monotonic density assumption might be questionable. While testing qualitative constraints such as monotonicity are difficult in general, we show that a likelihood ratio test statistic Kₙ has an incredibly simple asymptotic null distribution: n¹⸍²(Kₙ-γ), where γ is the Euler-Mascheroni constant, converges to a normal distribution with mean 0 and variance π²/6-1. The results are shown based on a connection between the test statistic and uniform spacing distributions, and by establishing a leading Oₚ(n-¹⸍⁶) remainder of the normalized test statistic.