  The two-dimensional infinite projected entangled pair states tensor network
is evolved in imaginary time with the full update (FU) algorithm to simulate
the Shastry-Sutherland model in a magnetic field at finite temperature directly
in the thermodynamic limit. We focus on the phase transition into the $m=1/2$
magnetization plateau, which was observed in experiments on
SrCu$_2$(BO$_3$)$_2$. For the largest simulated bond dimension, the early
evolution in the high-temperature regime is simulated with the simple update
(SU) scheme and then, as the correlation length increases, continued with the
FU scheme towards the critical regime. We apply a small symmetry-breaking bias
field and then extrapolate towards zero bias using a simple scaling theory in
the bias field. The combined SU + FU scheme provides an accurate estimate of
the critical temperature, even though the results could not be fully converged
in the bond dimension in the vicinity of the transition. The critical
temperature estimate is improved with a generalized scaling theory that
combines two divergent length scales: One due to the bias, and the other due to
the finite bond dimension. The obtained results are consistent with the
transition being in the universality class of the two-dimensional classical
Ising model. The estimated critical temperature is $3.5(2)$ K, which is well
above the temperature $2.1$ K used in the experiments.
