= Solution
Each spatial <eigenfunction> acquires the temporal factor $e^{-\kappa\lambda_{mn}t}$ under the <heat equation>. Expanding the initial constant using the preceding coefficients gives the <cooling of an initially uniform square>:
$$
\boxed{T(x,y;t)=\frac{16T_0}{\pi^2}\sum_{\substack{m,n\geq1\\m,n\text{ odd}}}\frac1{mn}\sin\frac{m\pi x}{\ell}\sin\frac{n\pi y}{\ell}\,e^{-\kappa\pi^2(m^2+n^2)t/\ell^2}.}
$$
For $t>0$ the exponential factors justify differentiating the series and enforce the zero boundary temperature. As $t\downarrow0$, it tends to $T_0$ at every interior point and in <L2 space>; the instantaneous boundary cooling is incompatible with continuity at the initial boundary corners, which is expected for this idealization.
The slowest decaying nonzero mode is $(1,1)$, giving
$$
\boxed{T(x,y;t)\sim\frac{16T_0}{\pi^2}\sin\frac{\pi x}{\ell}\sin\frac{\pi y}{\ell}\,e^{-2\kappa\pi^2t/\ell^2}}
$$
at each fixed interior point when $T_0\ne0$. The next contributing modes have $(m,n)=(1,3),(3,1)$ and decay as $e^{-10\kappa\pi^2t/\ell^2}$, making the leading-mode approximation exponentially accurate.
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