  We study the average density of $N$ spinless noninteracting fermions in a
$2d$ harmonic trap rotating with a constant frequency $\Omega$ and in the
presence of an additional repulsive central potential $\gamma/r^2$. The average
density at zero temperature was recently studied in Phys. Rev. A
$\textbf{103}$, 033321 (2021) and an interesting multi-layered "wedding cake"
structure with a "hole" at the center was found for the density in the large
$N$ limit. In this paper, we study the average density at finite temperature.
We demonstrate how this "wedding-cake" structure is modified at finite
temperature. These large $N$ results warrant going much beyond the standard
Local Density Approximation. We also generalize our results to a wide variety
of trapping potentials and demonstrate the universality of the associated
scaling functions both in the bulk and at the edges of the "wedding-cake".
