We present a high precision H-band emission spectrum of the transiting brown dwarf KELT-1b, which we spectrophotometrically observed during a single secondary eclipse using the LUCI1 multi-object spectrograph on the Large Binocular Telescope. Using a Gaussian-process regression model, we are able to clearly measure the broadband eclipse depth as Delta-H=1418+/-94ppm. We are also able to spectrally-resolve the H-band into five separate wavechannels and measure the eclipse spectrum of KELT-1b at R~50 with an average precision of +/-135ppm. We find that the day side has an average brightness temperature of 3250+/-50K, with significant variation as a function of wavelength. Based on our observations, and previous measurements of KELT-1b's eclipse at other wavelengths, we find that KELT-1b's day side appears identical to an isolated 3200K brown dwarf, and our modeling of the atmospheric emission shows a monotonically decreasing temperature-pressure profile. This is in contrast to hot Jupiters with similar day side brightness temperatures near 3000K, all of which appear to be either isothermal or posses a stratospheric temperature inversion. We hypothesize that the lack of an inversion in KELT-1b is due to its high surface gravity, which we argue could be caused by the increased efficiency of cold-trap processes within its atmosphere.