  [Abridged] Protoplanetary disks have been studied extensively, both
physically and chemically, to understand the environment in which planets form.
However, the first steps of planet formation are likely to occur already when
the protostar and disk are still embedded in their natal envelope. The initial
conditions for planet formation may thus be provided by these young embedded
disks, of which the physical and chemical structure is poorly characterized. We
aim to constrain the midplane temperature structure, one of the critical
unknowns, of the embedded disk around L1527. In particular, we set out to
determine whether there is an extended cold outer region where CO is frozen
out, as is the case for Class II disks. We use archival ALMA data to directly
observe the midplane of the near edge-on L1527 disk. Optically thick $^{13}$CO
($J=2-1$) and C$^{18}$O ($J=2-1$) emission is observed throughout the disk and
inner envelope, while N$_2$D$^+ (J=3-2$), which can only be abundant when CO is
frozen out, is not detected. Both CO isotopologues have brightness temperatures
$\gtrsim$ 25 K along the midplane. Disk and envelope emission can be
disentangled kinematically, because the largest velocities are reached in the
disk. A power law radial temperature profile constructed using the highest
midplane temperature at these velocities suggest that the temperature is above
20 K out to at least 75 AU, and possibly throughout the entire 125 AU disk.
Radiative transfer models show that a model without CO freeze-out in the disk
matches the C$^{18}$O observations better than a model with the CO snowline at
$\sim$70 AU. In addition, there is no evidence for a large (order of magnitude)
depletion of CO. The disk around L1527 is likely to be warm enough to have CO
present in the gas phase throughout the disk, suggesting that young embedded
disks can indeed be warmer than the more evolved Class II disks.
