A transition path is a part of a one-dimensional trajectory of a diffusing particle, which starts from point a and is terminated as soon as it comes to point b for the first time. It is the trajectory's final segment that leaves point a and goes to point b without returning to point a. The duration of this segment is called transition path time or, alternatively, direct transit time. We study the mean transition path time in monotonically increasing entropy potentials of the narrowing cones in spaces of different dimensions. We find that this time, normalized to its value in the absence of the potential, monotonically increases with the barrier height for the entropy potential of a narrowing two-dimensional cone, is independent of the barrier height for a narrowing three-dimensional cone, and monotonically decreases with the barrier height for narrowing cones in spaces of higher dimensions. Moreover, we show that as the barrier height tends to infinity, the normalized mean transition path time approaches its universal limiting value n/3, where n = 2, 3, 4, ... is the space dimension. This is in sharp contrast to the asymptotic behavior of this quantity in the case of a linear potential of mean force, for which it approaches zero in this limit.