Increase the BBOX size but keep the ratio (Lon/Lat)

Perhaps ST_Expand is what you're looking for. You could first calculate the area of the input bbox using ST_Area and then use the output as a unit to expand the bbox.

SELECT            -- here you can play with different sizes
  ST_Expand(geom, ST_Area(geom)/2) 
FROM yourtable;

Example:

WITH j (geom) AS (
  SELECT ST_MakeEnvelope(11.555333537980914,
                         47.76067947037518,
                         11.995692579075694,
                         48.281587762758136,4326)
)
SELECT 
  ST_Expand(geom,ST_Area(geom)/2) 
FROM j;

The image below represents the result set. The inner bbox is the one you provided and the outer one was created with ST_Expand.

Demo: db<>fiddle

The answer provided by @Jim Jones works perfectly. Is there something PostGIS can not do? :)

I did not wanted to be dependend on PostGIS

so I tried to solve the problem with R. My approach:

I prolong each diagonal of the bbox and calculate the bearing for that diagonal. Based on that data I calculate new edge points of the bbox. It kind of works but the left side of the bbox looks a bit small. I think there is a misstake somwhere but i dont know yet where.

xmin<- 11.555333537980914 
ymin<- 47.76067947037518 
xmax<- 11.995692579075694 
ymax<- 48.281587762758136 

###calculate bearing clockwise of diagonal for each corner of the BBOX
######## right bottom, left und top
######## left and bottom, right and top 
######## left and top and right and bottom 
######## right and top, left and bottom 
##bearing(p1, p2, a=6378137, f=1/298.257223563)
bearing1 <- geosphere::bearingRhumb(c(xmax,ymin),c(xmin,ymax))
bearing2 <- geosphere::bearingRhumb(c(xmin,ymin),c(xmax,ymax))
bearing3 <- geosphere::bearingRhumb(c(xmin,ymin),c(xmax,ymin))
bearing4 <- geosphere::bearingRhumb(c(xmax,ymax),c(xmin,ymin))
#new bbox points
########################## left und top
########################## right und top
########################## right und bottom
########################## left und bottom
p1<- geosphere::destPointRhumb(c(xmin,ymax), bearing1, 10000, r = 6378137)
p2<- geosphere::destPointRhumb(c(xmax,ymax), bearing2, 10000, r = 6378137)
p3<- geosphere::destPointRhumb(c(xmax,ymin), bearing3, 10000, r = 6378137)
p4<- geosphere::destPointRhumb(c(xmin,ymin), bearing4, 10000, r = 6378137)

data<-rbind.data.frame(p1,p2,p3,p4)

xmin<-min(data$lon)
ymin<-min(data$lat)
xmax<-max(data$lon)
ymax<-max(data$lat)
cat(xmin,",",ymin,",",xmax,",",ymax)

One solution would be to translate the box so that its center is at the origin, multiply everything with 1.5, and then translate back. This should be possible with a single ST_Affine(), but I'm too lazy to work out the details. :)