  We study the sandpile model on three-dimensional spanning Ising clusters with
the temperature $T$ treated as the control parameter. By analyzing the three
dimensional avalanches and their two-dimensional projections (which show
scale-invariant behavior for all temperatures), we uncover two universality
classes with different exponents (an ordinary BTW class, and SOC$_{T=\infty}$),
along with a tricritical point (at $T_c$, the critical temperature of the host)
between them. The transition between these two criticalities is induced by the
transition in the support. The SOC$_{T=\infty}$ universality class is
characterized by the exponent of the avalanche size distribution
$\tau^{T=\infty}=1.27\pm 0.03$, consistent with the exponent of the size
distribution of the Barkhausen avalanches in amorphous ferromagnets (Phys. Rev.
L 84, 4705 (2000)). The tricritical point is characterized by its own critical
exponents. In addition to the avalanche exponents, some other quantities like
the average height, the spanning avalanche probability (SAP) and the average
coordination number of the Ising clusters change significantly the behavior at
this point, and also exhibit power-law behavior in terms of $\epsilon\equiv
\frac{T-T_c}{T_c}$, defining further critical exponents. Importantly the finite
size analysis for the activity (number of topplings) per site shows the scaling
behavior with exponents $\beta=0.19\pm 0.02$ and $\nu=0.75\pm 0.05$. A similar
behavior is also seen for the SAP and the average avalanche height. The fractal
dimension of the external perimeter of the two-dimensional projections of
avalanches is shown to be robust against $T$ with the numerical value
$D_f=1.25\pm 0.01$.
