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 moiré pattern domains from the nonzero initial Novaco-McTague equilibrium angle to nearly zero. In the temperature (T ) -corrugation strength (W0) 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 = Tc, 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 = Tc. The experimental pursuit of this novel scenario is proposed.