  The Fermi bubbles are two lobes filled with non-thermal particles that emit
gamma rays, extend $\approx$10 kpc vertically from the Galactic center, and
formed from either nuclear star formation or accretion activity on Sgr A*.
Simulations predict a range of shock strengths as the bubbles expand into the
surrounding hot gas halo distribution ($T_{halo} \approx 2 \times 10^6$ K), but
with significant uncertainties in the energetics, age, and thermal gas
structure. The bubbles should contain thermal gas with temperatures between
$10^6$ and $10^8$ K, with potential X-ray signatures. In this work, we
constrain the bubbles' thermal gas structure by modeling the OVII and OVIII
emission line strengths from archival XMM-Newton and Suzaku data. Our emission
model includes a hot thermal volume-filled bubble component cospatial with the
gamma-ray region, and a shell of compressed material. We find that a
bubble/shell model with $n \approx 1 \times 10^{-3}$ cm$^{-3}$ and with
log($T$) $\approx$ 6.60-6.70 is consistent with the observed line intensities.
In the framework of a continuous Galactic outflow, we infer a bubble expansion
rate, age, and energy injection rate of $490_{-77}^{+230}$ km s$^{-1}$,
$4.3_{-1.4}^{+0.8}$ Myr, and $2.3_{-0.9}^{+5.1} \times 10^{42}$ erg s$^{-1}$.
These estimates are consistent with the bubbles forming from a Sgr A* accretion
event rather than from nuclear star formation.
