= Solution
For an ideally dissociated <electrolyte>, the <Debye–Hückel screening length> is
$$
\lambda_D=\left(\frac{\varepsilon_0\varepsilon_r k_BT}
{e^2\sum_i n_i z_i^2}\right)^{1/2},
\qquad I=\frac12\sum_i c_i z_i^2.
$$
Here $n_i$ is ionic <number density>, $z_i$ the charge in units of the <elementary charge>, and $c_i$ is molar concentration. Conversion between $I$ and <number density> cancels in a ratio. At fixed <relative permittivity>, $\lambda_D\propto\sqrt{T/I}$.
In $0.01\,\mathrm M$ <sodium chloride>, $c_{\rm Na^+}=c_{\rm Cl^-}=0.01\,\mathrm M$, giving $I_1=0.01\,\mathrm M$. Complete dissociation of $0.1\,\mathrm M$ <aluminium chloride> gives $c_{\rm Al^{3+}}=0.1\,\mathrm M$ and $c_{\rm Cl^-}=0.3\,\mathrm M$, so
$$
I_2=\frac12[0.1(3)^2+0.3(1)^2]=0.60\,\mathrm M=60I_1.
$$
The factor is sixty, not ten: the squared charge and the three counterions per formula unit both matter. Using the given $5\,\mathrm{nm}$ reference length directly, and $T_1\simeq298\,\mathrm K$, $T_2=323\,\mathrm K$,
$$
\boxed{\lambda_{D,2}=5\,\mathrm{nm}\sqrt{\frac{323}{298}\frac{0.01}{0.60}}
\simeq0.67\,\mathrm{nm}.}
$$
The modest <temperature> increase raises the screening length by only about $4\%$ at fixed concentration; the large increase in <ionic strength> shortens it by almost an order of magnitude overall.
Under <charge stabilization of colloids>, the electrostatic double-layer repulsion now has a much shorter range. Attractive <Van der Waals forces> can then dominate at closer separations, reducing the barrier to <particle aggregation> in <DLVO theory>. Thus \b[an electrostatically stabilized <colloid> is likely to be destabilized], with aggregation more likely than in the original dispersion.
This is a conditional physical prediction, not a proof of instability from a length alone. No particle radius, surface potential, attractive interaction strength or steric coating is specified. <Steric stabilization> could preserve stability, and trivalent-ion <adsorption> can alter the surface charge. Moreover, ideal screening theory is only an estimate at $I\simeq0.6\,\mathrm M$, where ion association, <hydrolysis> and activities can matter. The numerical result uses the supplied reference length and the stipulated unchanged permittivity, rather than replacing either with an unprovided calibration.
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