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A geometrized luminosity is dimensionless because in geometrized units an energy and a time have the same dimension. Starting from SI luminosity, multiply by $G/c^5$. The corresponding SI power unit is $c^5/G$. Radiating a fraction $f$ of a mass $M$ at constant SI luminosity $L$ takes $fM{c}^2/L$, or $f(GM/c^3)/L_{\mathrm{geom}}$ in seconds.

Power is energy per time. In geometrized units, energy has the dimension of length and time is converted to length using $c$, so the dimensionless geometrized luminosity isEquivalently, $c^5/G$ is the corresponding SI power unit. The lifetime for emitting the entire rest energy at the given constant luminosity isThe same result follows from $M_{\odot }^{(\mathrm{time})}/L_{\odot ,\mathrm{geom}}$. The stipulated full-mass radiation time is of order $10^{13}$ years; this is the constant-luminosity energy budget asked for, not a stellar-evolution calculation.