  Solid-state emitters are excellent candidates for developing integrated
sources of single photons. Yet, phonons degrade the photon indistinguishability
both through pure dephasing of the zero-phonon line and through phonon-assisted
emission. Here, we study theoretically and experimentally the
indistinguishability of photons emitted by a semiconductor quantum dot in a
microcavity as a function of temperature. We show that a large coupling to a
high quality factor cavity can simultaneously reduce the effect of both
phonon-induced sources of decoherence. It first limits the effect of pure
dephasing on the zero phonon line with indistinguishabilities above $97\%$ up
to $18$ K. Moreover, it efficiently redirects the phonon sidebands into the
zero-phonon line and brings the indistinguishability of the full emission
spectrum from $87\%$ (resp. $24\%$) without cavity effect to more than $99\%$
(resp. $76\%$) at $0$ K (resp. $20$ K). We provide guidelines for optimal
cavity designs that further minimize the phonon-induced decoherence.
