  We have performed large-scale Monte Carlo simulations on a two-dimensional
generalized Ashkin-Teller model to calculate the thermodynamic properties in
the critical region near its transitions. The Ashkin-Teller model has a pair of
Ising spins at each site which interact with neighboring spins through
pair-wise and 4-spin interactions. The model represents the interactions
between orbital current loops in $Cu O_2$-plaquettes of high-$T_c$ cuprates,
which order with a staggered magnetization $\Mso$ inside each unit-cell in the
underdoped region of the phase diagram below a temperature $T^*(x)$ which
depends on doping. The pair of Ising spins per unit-cell represent the
directions of the currents in the links of the current loops. The
generalizations are the inclusion of anisotropy in the pair-wise nearest
neighbor current-current couplings consistent with the symmetries of a square
lattice and the next nearest neighbor pair-wise couplings. We use the Binder
cumulant to estimate the correlation length exponent $\nu$ and the order
parameter exponent $\beta$. Our principal results are that in a range of
parameters, the Ashkin-Teller model as well as its generalization has an order
parameter susceptibility which diverges as $T \to T^*$ and an order parameter
below $T^*$. Importantly, however, there is no divergence in the specific heat.
This puts the properties of the model in accord with the experimental results
in the underdoped cuprates. We also calculate the magnitude of the "bump" in
the specific heat in the critical region to put limits on its observability.
Finally, we show that the staggered magnetization couples to the uniform
magnetization $M_0$ such that the latter has a weak singularity at $T^*$ and
also displays a wide critical region, also in accord with recent experiments.
