Recent advances in microelectronics and biomedical technology has led to capabilities for integrating programmable cooling mechanisms into bioelectronic devices for pain management and other important applications in human health. However, achieving localized and precise targeted cooling remains a challenge due to the influence of tissue mechanics during device placement/operation, viscous effects in the flow of coolants, and critical physical parameters that locally affect tissue temperature and mechanical responses. Recent studies demonstrate that bioelectronic devices can be equipped with microfluidic structures designed to support phase-change cooling mechanisms, with cooling power adjustable by control of fluid flow rates. When operated in closed-loop feedback schemes based on measurements of temperature at the tissue interface, these platforms can achieve precise temperature control of regions of peripheral nerves involved in pain signaling. Establishing a theoretical model to understand the effects of these devices on the underlying soft and deformable tissues, along with microfluidic deformations that can alter the heat transfer cooling process in viscous flow, is critical for optimized design and operation. To address this, a theoretical model is developed to describe the influence of remote strain in the tissue and to quantify the spatial temperature distribution of tissue cooled by fluid phase change in deformable microfluidic channels. A scaling law is derived to quantify the steady-state temperature distribution in the tissue under physiologically relevant mechanical loads, fluid viscosity, flow rates, and thermal conductivities, enabling the prediction of liquid and gas flux based on expected temperature changes and saturated vapor pressure in the deformed tissue and microchannel. These findings reveal that within a reasonable physiological range, the influence of fluid viscosity on temperature changes can be neglected, and the ratio of fluid to tissue thermal conductivity minimally impacts the temperature change at relatively high flow rates. Additionally, the model suggests a limit to tissue cooling for a given fluid, which may not necessarily increase with higher fluid volumes.