  We present the ultraviolet luminosity function and an estimate of the cosmic
star formation rate density at $8<z<13$ derived from deep NIRCam observations
taken in parallel with the MIRI Deep Survey (MDS) of the Hubble Ultra Deep
Field (HUDF), NIRCam covering the parallel field 2 (HUDF-P2). Our deep (40
hours) NIRCam observations reach a F277W magnitude of 30.8 ($5\sigma$), more
than 2 magnitudes deeper than JWST public datasets already analyzed to find
high redshift galaxies. We select a sample of 44 $z>8$ galaxy candidates based
on their dropout nature in the F115W and/or F150W filters, a high probability
for their photometric redshifts, estimated with three different codes, being at
$z>8$, good fits based on $\chi^2$ calculations, and predominant solutions
compared to $z<8$ alternatives. We find mild evolution in the luminosity
function from $z\sim13$ to $z\sim8$, i.e., only a small increase in the average
number density of $\sim$0.2 dex, while the faint-end slope and absolute
magnitude of the knee remain approximately constant, with values
$\alpha=-2.2\pm0.1$ and $M^*=-20.8\pm0.2$ mag. Comparing our results with the
predictions of state-of-the-art galaxy evolution models, we find two main
results: (1) a slower increase with time in the cosmic star formation rate
density compared to a steeper rise predicted by models; (2) nearly a factor of
10 higher star formation activity concentrated in scales around 2 kpc in
galaxies with stellar masses $\sim10^8$ M$_\odot$ during the first 350 Myr of
the Universe, $z\sim12$, with models matching better the luminosity density
observational estimations $\sim$150 Myr later, by $z\sim9$.
