  The Aubry unpinned--pinned transition in the sliding of two incommensurate
lattices occurs for increasing mutual interaction strength in one dimension
($1D$) and is of second order at $T=0$, turning into a crossover at nonzero
temperatures. Yet, real incommensurate lattices come into contact in two
dimensions ($2D$), at finite temperature, generally developing a mutual
Novaco-McTague misalignment, conditions in which the existence of a sharp
transition is not clear. Using a model inspired by colloid monolayers in an
optical lattice as a test $2D$ case, simulations show a sharp Aubry transition
between an unpinned and a pinned phase as a function of corrugation. Unlike
$1D$, the $2D$ transition is now of first order, and, importantly, remains well
defined at $T>0$. It is heavily structural, with a local rotation of moir\'e
pattern domains from the nonzero initial Novaco-McTague equilibrium angle to
nearly zero. In the temperature ($T$) -- corrugation strength ($W_0$) plane,
the thermodynamical coexistence line between the unpinned and the pinned phases
is strongly oblique, showing that the former has the largest entropy. This
first-order Aubry line terminates with a novel critical point $T=T_c$, marked
by a susceptibility peak. The expected static sliding friction upswing between
the unpinned and the pinned phase decreases and disappears upon heating from
$T=0$ to $T=T_c$. The experimental pursuit of this novel scenario is proposed.
