  Renormalization group flows are studied between 5d SCFTs engineered by
$(p,q)$ 5-brane webs with large numbers of external 5-branes. A general
expression for the free energy on $S^5$ in terms of single-valued trilogarithm
functions is derived from their supergravity duals, which are characterized by
the 5-brane charges and additional geometric parameters. The additional
geometric parameters are fixed by regularity conditions, and we show that the
solutions to the regularity conditions extremize a trial free energy. These
results are used to survey a large sample of $\mathcal O(10^5)$ renormalization
group flows between different 5d SCFTs, including Higgs branch flows and flows
that preserve the $SU(2)$ $R$-symmetry. In all cases the free energy changes
monotonically towards the infrared, in line with a 5d $F$-theorem.
