  Originally proposed as a cosmological probe of the large-scale structure,
line intensity mapping (LIM) also offers a unique window into the astrophysics
of galaxy evolution. Adding to the astrophysical explorations of LIM technique
that have traditionally focused on small, non-linear scales, we present a novel
method to study the global star formation law using forthcoming data from
large-scale baryonic acoustic oscillation (BAO) intensity mapping. Using the
amplitude of the percent-level but scale-dependent bias induced by baryon
fraction fluctuations on BAO scales, we show that combining auto- and
cross-correlation power spectra of two (or more) LIM signals allows to probe
the star formation law power index $\mathcal{N}$. We examine the prospect for
mapping H$\alpha$ and [OIII] lines across all scales, especially where imprints
of the baryon fraction deviation exist, with space missions like SPHEREx. We
show that although SPHEREx may only marginally probe $\mathcal{N}$ by accessing
a modest number of large-scale modes in its 200 deg$^2$ deep survey, future
infrared all-sky surveys reaching a comparable depth with an improved spectral
resolution ($R \gtrsim 400$) are likely to constrain $\mathcal{N}$ to a
precision of 10$-$30%, sufficient for distinguishing models with varying
feedback assumptions, out to $z\sim4$ using BAO intensity mapping. Leveraging
this effect, large, cosmic-variance-limited LIM surveys in the far future can
scrutinize the physical connection between galaxy evolution and the large-scale
cosmological environment, while performing stringent tests of the standard
cosmological model.
