The exact nature of the luminous fast blue optical transient AT 2018cow is still debated. In this first of a two-paper series, we present a detailed analysis of three Hubble Space Telescope (HST) observations of AT 2018cow covering ∼ 50–60 days post-discovery in combination with other observations throughout the first two months, and derive significantly improved constraints of the late thermal properties. By modeling the spectral energy distributions (SEDs), we confirm that the UV-optical emission over 50–60 days was still a smooth blackbody (i.e., optically thick) with a high temperature (TBB ∼ 15000 K) and small radius (RBB ≲ 1000 R⊙). Additionally, we report for the first time a break in the bolometric light curve: the thermal luminosity initially declined at a rate of LBB ∝ t−2.40, but faded much faster at t−3.06 after day 13. Reexamining possible late-time power sources, we disfavor significant contributions from radioactive decay based on the required 56Ni mass and lack of UV line blanketing in the HST SEDs. We argue that the commonly proposed interaction with circumstellar material may face significant challenges in explaining the late thermal properties, particularly the effects of the optical depth. Alternatively, we find that continuous outflow/wind driven by a central engine can still reasonably explain the combination of a receding photosphere, optically thick and rapidly fading emission, and intermediate-width lines. However, the rapid fading may have further implications on the power output and structure of the system. Our findings may support the hypothesis that AT 2018cow and other “Cow-like transients” are powered mainly by accretion onto a central engine.