  We investigate the evolution over the last 6.3 Gyr of cosmic time (i.e.,
since z ~ 0.7) of the average X-ray properties of early-type galaxies within
the Extended Chandra Deep Field-South (E-CDF-S). Our early-type galaxy sample
includes 539 objects with red-sequence colors and Sersic indices larger than n
= 2.5, which were selected jointly from the COMBO-17 (Classifying Objects by
Medium-Band Observations in 17 Filters) and GEMS (Galaxy Evolution from
Morphologies and SEDs) surveys. We utilize the deep Chandra observations over
the E-CDF-S and X-ray stacking analyses to constrain primarily the average
X-ray emission from "normal" early-type galaxies (i.e., those that are not
dominated by luminous active galactic nuclei [AGNs]). In our analyses, we study
separately optically luminous (L_B ~ 10^[10-11] L_B,sol) and faint (L_B ~
10^[9.3-10] L_B,sol) galaxy samples, which we expect to have soft (0.5-2.0 keV)
X-ray emission dominated by hot (~1 keV) interstellar gas and low mass X-ray
binary (LMXB) populations, respectively. We detect individually 49 (~9%) of our
galaxies in the X-ray band, and classify these sources as either normal
early-type galaxies (17 galaxies) or AGN candidates (32 galaxies). The AGN
fraction of our optically luminous samples evolves with redshift in a manner
consistent with the (1+z)^3 evolution observed in other investigations of
X-ray-selected AGNs. After removing potential AGNs from our samples, we find
that the X-ray-to-B-band mean luminosity ratio (L_X/L_B) for optically luminous
early-type galaxies does not evolve significantly over the redshift range
z~0.0-0.7. This lack of X-ray evolution implies a general balance between the
heating and cooling of the hot interstellar gas [Abridged].
