  Integrated optomechanical systems are one of the leading platforms for
manipulating, sensing, and distributing quantum information. The temperature
increase due to residual optical absorption sets the ultimate limit on
performance for these applications. In this work, we demonstrate a
two-dimensional optomechanical crystal geometry, named \textbf{b-dagger}, that
alleviates this problem through increased thermal anchoring to the surrounding
material. Our mechanical mode operates at 7.4 GHz, well within the operation
range of standard cryogenic microwave hardware and piezoelectric transducers.
The enhanced thermalization combined with the large optomechanical coupling
rates, $g_0/2\pi \approx 880~\mathrm{kHz}$, and high optical quality factors,
$Q_\text{opt} = 2.4 \times 10^5$, enables the ground-state cooling of the
acoustic mode to phononic occupancies as low as $n_\text{m} = 0.35$ from an
initial temperature of 3 kelvin, as well as entering the optomechanical
strong-coupling regime. Finally, we perform pulsed sideband asymmetry of our
devices at a temperature below 10 millikelvin and demonstrate ground-state
operation ($n_\text{m} < 0.45$) for repetition rates as high as 3 MHz. Our
results extend the boundaries of optomechanical system capabilities and
establish a robust foundation for the next generation of microwave-to-optical
transducers with entanglement rates overcoming the decoherence rates of
state-of-the-art superconducting qubits.
