  Numerical simulations of heat transport in diamond anvil cells reveal a
possibility for absolute measurements of specific heat via high-frequency
modulation calorimetry. Such experiments could reveal and help characterize
temperature-driven phase transitions at high-pressure, such as melting, the
glass transition, magnetic and electric orderings, or superconducting
transitions. Specifically, we show that calorimetric information of a sample
cannot be directly extracted from measurements at frequencies slower than the
timescale of conduction to the diamond anvils (10s to 100s of kHz) since the
experiment is far from adiabatic. At higher frequencies, laser-heating
experiments allow relative calorimetric measurements, where changes in specific
heat of the sample are discriminated from changes in other material properties
by scanning the heating frequency from $\sim 1$ MHz to 1 GHz. But laser-heating
generates large temperature gradients in metal samples, preventing absolute
heat capacities to be inferred. High-frequency Joule heating, on the other
hand, allows accurate, absolute specific heat measurements if it can be
performed at high-enough frequency: assuming a thin layer of KBr insulation,
the specific heat of a 5 $\mu$m-thick metal sample heated at 100 kHz, 1 MHz, or
10 MHz frequency would be measured with 30%, 8% or 2% accuracy, respectively.
