There are a couple of ways you could approach this, and in any event, you don't need to have the same cell size for input rasters (DEM and temperature grid). What you want to pay attention to with Raster Calculator is the Environments settings. Specifically, the Output Coordinates information (for reference system and units of output raster), the Processing Extent (the area you want represented), and most importantly the Raster Analysis - in these drop-down options, you can set your cell size for the output raster. The default is maximum of inputs, which means whichever raster layer has the coarsest resolution will be set for your output (if you don't set anything).
Let's say your temperature raster has 30 meter resolution, and your DEM has 10 meter resolution (for example, it might be reversed). If set the "Raster Analysis" Cell Size to 10 in Environments, and perform a calculation, that calculation will be performed for every DEM cell and respective value, against the temperature value for the larger cell. Your output raster will be 10 meters, and each 30 meter temperature value will have been computed with 9 different elevations.
The calculator doesn't require matching resolutions, it simply works based on XY overlap, and for each cell in the output what values are represented in that location by the respective input layers.
As a complete alternative, if you want you can create vector points, and run an Extract Multi Values tool - this would allow you to create attributes for each point based on multiple rasters at the same time. But I think calculator sounds like what you want.