The persistent $H_0$ and $S_8$ tensions motivate the search for new dark-energy mechanisms capable of modifying the late-time expansion history while preserving the successful early-Universe predictions of $\Lambda$CDM scenario. Wald-Gauss-Bonnet (WGB) topological dark energy provides a physically motivated realization of this possibility, where the effective dark-energy sector emerges from cosmic horizon thermodynamics and the black-hole formation and merger history. We present the first early- and late-Universe analysis of WGB cosmology, implementing the model as an effective fluid in a modified solver and constraining it against CMB data from , ACT~DR6 and SPT-3G, DESI~DR2 BAO, and Pantheon+ supernovae. While late-time data alone are consistent with $\Lambda$CDM, the full dataset prefers a non-zero WGB contribution, $\Cn=0.387^{+0.118}_{-0.072}$, corresponding to a $>3\sigma$ phantom-like deviation and an improved fit ($\Delta {\chi }^2=-10.3$). The preferred solution raises $H_0$ from $68.5$ to $69.8~\kmsmpc$, reducing the Hubble tension by $\sim 1\sigma$, at the cost of a mild increase in $S_8$. The reconstructed cosmological observables show that WGB leaves the primary CMB almost unchanged while enhancing lensing and small-scale clustering, revealing a characteristic $H_0$-$S_8$ trade-off. WGB dark energy therefore emerges as a physically motivated and observationally viable late-time mechanism for partially alleviating the Hubble tension without introducing new early-Universe physics.