  This is the third paper in a series analyzing X-ray emission from the hot
interstellar medium in a sample of 54 normal elliptical galaxies observed by
Chandra, focusing on 36 galaxies with sufficient signal to compute radial
temperature profiles. We distinguish four qualitatively different types of
profile: positive gradient (outwardly rising), negative gradients (falling),
quasi-isothermal (flat) and hybrid (falling at small radii, then rising). We
measure the mean logarithmic temperature gradients in two radial regions: from
0--2 $J$-band effective radii $R_J$ (excluding the central point source), and
from 2--$4R_J$. We find the outer gradient to be uncorrelated with intrinsic
host galaxy properties, but strongly influenced by the environment: galaxies in
low-density environments tend to show negative outer gradients, while those in
high-density environments show positive outer gradients, suggesting influence
of circumgalactic hot gas. The inner temperature gradient is unaffected by the
environment but strongly correlated with intrinsic host galaxy characteristics:
negative inner gradients are more common for smaller, optically faint, low
radio-luminosity galaxies, whereas positive gradients are found in bright
galaxies with stronger radio sources. There is no evidence for bimodality in
the distribution of inner or outer gradients. We propose three scenarios to
explain the inner temperature gradients: (1) Weak AGN heat the ISM locally,
higher-luminosity AGN heat the system globally through jets inflating cavities
at larger radii; (2) The onset of negative inner gradients indicates a
declining importance of AGN heating relative to other sources, such as
compressional heating or supernovae; (3) The variety of temperature profiles
are snapshots of different stages of a time-dependent flow.
